%!PS-Adobe-3.0 %%Title: (Microsoft Word - equivalence5) %%Creator: (Microsoft Word: LaserWriter 8 8.3.4) %%CreationDate: (10:34 PM Tuesday, March 25, 1997) %%For: (peter) %%Pages: 12 %%DocumentFonts: Times-Roman Symbol Times-Bold Times-Italic %%DocumentNeededFonts: Times-Roman Symbol Times-Bold Times-Italic %%DocumentSuppliedFonts: %%DocumentData: Clean7Bit %%PageOrder: Ascend %%Orientation: Portrait %%DocumentMedia: Default 612 792 0 () () %ADO_ImageableArea: 31 31 583 761 %%EndComments userdict begin/dscInfo 5 dict dup begin /Title(Microsoft Word - equivalence5)def /Creator(Microsoft Word: LaserWriter 8 8.3.4)def /CreationDate(10:34 PM Tuesday, March 25, 1997)def /For(peter)def /Pages 12 def end def end save /version23-manualfeedpatch where { pop false } { true }ifelse % we don't do an explicit 'get' since product and version MAY % be in systemdict or statusdict - this technique gets the lookup % without failure statusdict begin product (LaserWriter) eq % true if LaserWriter version cvr 23.0 eq % true if version 23 end and % only install this patch if both are true and % true only if patch is not installed and is for this printer % save object and boolean on stack dup { exch restore }if % either true OR saveobject false dup { /version23-manualfeedpatch true def /oldversion23-showpage /showpage load def /showpage % this showpage will wait extra time if manualfeed is true {% statusdict /manualfeed known {% manualfeed known in statusdict statusdict /manualfeed get {% if true then we loop for 5 seconds usertime 5000 add % target usertime { % loop dup usertime sub 0 lt { exit }if }loop pop % pop the usertime off the stac }if }if oldversion23-showpage }bind def }if not{ restore }if /md 202 dict def md begin/currentpacking where {pop /sc_oldpacking currentpacking def true setpacking}if %%BeginFile: adobe_psp_basic %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. All Rights Reserved. /bd{bind def}bind def /xdf{exch def}bd /xs{exch store}bd /ld{load def}bd /Z{0 def}bd /T/true /F/false /:L/lineto /lw/setlinewidth /:M/moveto /rl/rlineto /rm/rmoveto /:C/curveto /:T/translate /:K/closepath /:mf/makefont /gS/gsave /gR/grestore /np/newpath 14{ld}repeat /$m matrix def /av 83 def /por true def /normland false def /psb-nosave{}bd /pse-nosave{}bd /us Z /psb{/us save store}bd /pse{us restore}bd /level2 /languagelevel where { pop languagelevel 2 ge }{ false }ifelse def /featurecleanup { stopped cleartomark countdictstack exch sub dup 0 gt { {end}repeat }{ pop }ifelse }bd /noload Z /startnoload { {/noload save store}if }bd /endnoload { {noload restore}if }bd level2 startnoload /setjob { statusdict/jobname 3 -1 roll put }bd /setcopies { userdict/#copies 3 -1 roll put }bd level2 endnoload level2 not startnoload /setjob { 1 dict begin/JobName xdf currentdict end setuserparams }bd /setcopies { 1 dict begin/NumCopies xdf currentdict end setpagedevice }bd level2 not endnoload /pm Z /mT Z /sD Z /realshowpage Z /initializepage { /pm save store mT concat }bd /endp { pm restore showpage }def /$c/DeviceRGB def /rectclip where { pop/rC/rectclip ld }{ /rC { np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K clip np }bd }ifelse /rectfill where { pop/rF/rectfill ld }{ /rF { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl fill gR }bd }ifelse /rectstroke where { pop/rS/rectstroke ld }{ /rS { gS np 4 2 roll :M 1 index 0 rl 0 exch rl neg 0 rl :K stroke gR }bd }ifelse %%EndFile %%BeginFile: adobe_psp_colorspace_level1 %%Copyright: Copyright 1991-1993 Adobe Systems Incorporated. All Rights Reserved. /G/setgray ld /:F1/setgray ld /:F/setrgbcolor ld /:F4/setcmykcolor where { pop /setcmykcolor ld }{ { 3 { dup 3 -1 roll add dup 1 gt{pop 1}if 1 exch sub 4 1 roll }repeat pop setrgbcolor }bd }ifelse /:Fx { counttomark {0{G}0{:F}{:F4}} exch get exec pop }bd /:rg{/DeviceRGB :ss}bd /:sc{$cs :ss}bd /:dc{/$cs xdf}bd /:sgl{}def /:dr{}bd /:fCRD{pop}bd /:ckcs{}bd /:ss{/$c xdf}bd /$cs Z %%EndFile %%BeginFile: adobe_psp_uniform_graphics %%Copyright: Copyright 1990-1993 Adobe Systems Incorporated. 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page: 1 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (1)S gR gS 0 0 552 730 rC 80 54 :M f2_12 sf 2.242 .224(A Polynomial Time Algorithm For Determining DAG Equivalence in the)J 139 66 :M 2.554 .255(Presence of Latent Variables and Selection Bias)J 87 81 :M f0_10 sf -.009(by Peter Spirtes \(Department of Philosophy, 好色先生TV, ps7z@andrew.cmu.edu\))A 116 96 :M .009 .001( and Thomas Richardson \(Department of Statistics, University of Washington\))J 59 126 :M .146 .015(Following the terminology of Lauritzen et. al. \(1990\) say that a probability measure over )J 421 126 :M .056 .006(a )J 429 126 :M .303 .03(set )J 444 126 :M .144 .014(of )J 456 126 :M -.136(variables)A 59 138 :M f2_10 sf .41(V)A f0_10 sf .656 .066( satisfies the )J f2_10 sf 1.098 .11(local directed Markov property )J f0_10 sf .794 .079(for a directed acyclic )J 359 138 :M -.053(graph )A 385 138 :M -.163(\(DAG\) )A 416 138 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 427 138 :M .517 .052(with )J 449 138 :M -.074(vertices )A 483 138 :M f2_10 sf (V)S 59 150 :M f0_10 sf .099 .01(if and only if for every )J f4_10 sf .075(W)A f0_10 sf .048 .005( in )J f2_10 sf .065(V)A f0_10 sf (, )S f4_10 sf .075(W)A f0_10 sf .134 .013( is independent of)J f2_10 sf ( )S f0_10 sf .131 .013(the set of all its non-descendants conditional on the )J 480 150 :M -.053(set)A 59 162 :M .057 .006(of its parents. One natural question )J 203 162 :M .361 .036(that )J 222 162 :M -.044(arises )A 248 162 :M .517 .052(with )J 270 162 :M -.116(respect )A 301 162 :M .601 .06(to )J 313 162 :M (DAGs )S 342 162 :M .694 .069(is )J 353 162 :M -.039(when )A 378 162 :M .387 .039(two )J 397 162 :M (DAGs )S 426 162 :M -.235(are )A 441 162 :M (\322statistically)S 59 174 :M -.091(equivalent\323. )A 110 174 :M -.053(One )A 130 174 :M .139 .014(interesting )J 176 174 :M -.031(sense )A 202 174 :M .144 .014(of )J 215 174 :M .135 .014(\322statistical )J 262 174 :M -.224(equivalence\323 )A 316 174 :M .694 .069(is )J 328 174 :M -.181(\322d-separation )A 384 174 :M -.224(equivalence\323 )A 438 174 :M -.17(\(explained )A 483 174 :M .222(in)A 59 186 :M -.004(more detail below.\) In the case of DAGs, )A 227 186 :M -.16(d-separation )A 278 186 :M -.204(equivalence )A 327 186 :M .694 .069(is )J 338 186 :M .281 .028(also )J 358 186 :M -.165(corresponds )A 408 186 :M .601 .06(to )J 420 186 :M .056 .006(a )J 428 186 :M -.037(variety )A 459 186 :M .144 .014(of )J 471 186 :M -.136(other)A 59 198 :M -.037(natural )A 91 198 :M (senses )S 121 198 :M .144 .014(of )J 134 198 :M .35 .035(statistical )J 177 198 :M -.204(equivalence )A 227 198 :M -.031(\(such )A 253 198 :M .144 .014(as )J 266 198 :M -.076(representing )A 319 198 :M .218 .022(the )J 336 198 :M (same )S 361 198 :M .303 .03(set )J 377 198 :M .144 .014(of )J 390 198 :M .27 .027(distributions\). )J 452 198 :M -.14(Theorems)A 59 210 :M -.151(characterizing )A 118 210 :M -.16(d-separation )A 170 210 :M -.204(equivalence )A 220 210 :M -.052(for )A 236 210 :M -.337(directed )A 270 210 :M -.116(acyclic )A 302 210 :M -.026(graphs )A 333 210 :M -.313(and )A 351 210 :M .361 .036(that )J 371 210 :M -.126(can )A 389 210 :M .051 .005(be )J 403 210 :M -.207(used )A 425 210 :M .144 .014(as )J 438 210 :M .218 .022(the )J 455 210 :M .337 .034(basis )J 480 210 :M -.328(for)A 59 222 :M .474 .047(polynomial )J 110 222 :M .519 .052(time )J 133 222 :M .366 .037(algorithms )J 181 222 :M -.052(for )A 197 222 :M -.074(checking )A 237 222 :M -.16(d-separation )A 290 222 :M -.204(equivalence )A 341 222 :M -.231(were )A 365 222 :M -.256(provided )A 404 222 :M .417 .042(by )J 420 222 :M -.14(Verma )A 452 222 :M -.313(and )A 471 222 :M -.135(Pearl)A 59 234 :M -.015(\(1990\), and Frydenberg \(1990\). The question we will examine is how to extend these )A 402 234 :M .235 .023(results )J 432 234 :M .601 .06(to )J 444 234 :M -.119(cases )A 468 234 :M -.356(where)A 59 246 :M .056 .006(a )J 67 246 :M -.053(DAG )A 92 246 :M .218 .022(may )J 113 246 :M -.094(have )A 135 246 :M .18 .018(latent )J 161 246 :M -.205(\(unmeasured\) )A 217 246 :M -.065(variables )A 256 246 :M .144 .014(or )J 268 246 :M (selection )S 307 246 :M .281 .028(bias )J 327 246 :M .637 .064(\(i.e. )J 347 246 :M .281 .028(some )J 372 246 :M .144 .014(of )J 384 246 :M .218 .022(the )J 400 246 :M -.065(variables )A 440 246 :M .601 .06(in )J 453 246 :M .218 .022(the )J 470 246 :M -.329(DAG)A 59 258 :M -.084(have been )A 102 258 :M -.156(conditioned )A 151 258 :M .48 .048(on.\) )J 171 258 :M -.095(D-separation )A 225 258 :M -.204(equivalence )A 274 258 :M .694 .069(is )J 285 258 :M .144 .014(of )J 297 258 :M .04 .004(interest )J 330 258 :M .601 .06(in )J 342 258 :M (part )S 361 258 :M -.163(because )A 395 258 :M -.097(there )A 418 258 :M -.235(are )A 433 258 :M .366 .037(algorithms )J 480 258 :M -.328(for)A 59 270 :M .032 .003(constructing )J 114 270 :M (DAGs )S 145 270 :M .517 .052(with )J 169 270 :M .18 .018(latent )J 197 270 :M -.065(variables )A 238 270 :M -.313(and )A 257 270 :M (selection )S 298 270 :M .281 .028(bias )J 321 270 :M .361 .036(that )J 343 270 :M -.235(are )A 361 270 :M -.253(based )A 389 270 :M .417 .042(on )J 406 270 :M -.199(observed )A 447 270 :M -.099(conditional)A 59 282 :M .029 .003(independence relations. For this class of algorithms, it is impossible to determine which of two d-separation)J 59 294 :M -.115(equivalent )A 103 294 :M -.081(causal )A 131 294 :M -.037(structures )A 173 294 :M -.262(generated )A 213 294 :M .056 .006(a )J 221 294 :M .189 .019(given )J 248 294 :M .295 .03(probability )J 297 294 :M .387 .039(distribution, )J 351 294 :M .189 .019(given )J 378 294 :M .515 .052(only )J 401 294 :M .218 .022(the )J 418 294 :M .303 .03(set )J 434 294 :M .144 .014(of )J 447 294 :M -.099(conditional)A 59 306 :M -.192(independence and dependence relations true )A 231 306 :M .144 .014(of )J 243 306 :M .218 .022(the )J 259 306 :M -.199(observed )A 297 306 :M .387 .039(distribution. )J 350 306 :M -.188(We )A 367 306 :M .676 .068(will )J 387 306 :M -.226(describe )A 422 306 :M .056 .006(a )J 430 306 :M .474 .047(polynomial )J 480 306 :M -.053(\(in)A 59 318 :M -.002(the number of vertices\) time algorithm for determining when two DAGs which may have latent variables )A 483 318 :M -.328(or)A 59 330 :M -.081(selection bias are d-separation equivalent.)A 59 345 :M 1.349 .135(A DAG )J f4_10 sf .748(G)A f0_10 sf .235 .024( )J f2_10 sf 1.922 .192(entails a)J f0_10 sf .235 .024( )J f2_10 sf 2.377 .238(conditional independence relation )J f0_10 sf .927 .093(if and only if it is )J 391 345 :M (true )S 410 345 :M .601 .06(in )J 422 345 :M -.141(every )A 447 345 :M .012(probability)A 59 357 :M -.061(measure satisfying the local directed Markov )A 240 357 :M -.046(property )A 277 357 :M -.052(for )A 292 357 :M f4_10 sf .461(G)A f0_10 sf .29 .029(. )J 306 357 :M -.235(\(We )A 326 357 :M -.119(place )A 350 357 :M -.024(definitions )A 396 357 :M -.313(and )A 413 357 :M .361 .036(sets )J 432 357 :M .144 .014(of )J 444 357 :M -.065(variables )A 483 357 :M .222(in)A 59 369 :M -.034(boldface.\) Pearl, Geiger, and Verma \(Pearl 1988\) have shown that there is )A 356 369 :M .056 .006(a )J 364 369 :M -.078(graphical )A 404 369 :M .242 .024(relation, )J 441 369 :M -.16(d-separation,)A 59 381 :M .207 .021(that holds among three disjoint sets of variable )J f2_10 sf .102(A)A f0_10 sf .119 .012(, and )J f2_10 sf .094(B)A f0_10 sf .119 .012(, and )J f2_10 sf .102(C)A f0_10 sf .137 .014( in DAG )J f4_10 sf .102(G)A f0_10 sf .068 .007( if )J 374 381 :M -.313(and )A 391 381 :M .515 .052(only )J 413 381 :M .328 .033(if )J 423 381 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 434 381 :M .235 .023(entails )J 464 381 :M .361 .036(that )J 483 381 :M f2_10 sf (A)S 59 393 :M f0_10 sf .518 .052(is independent of )J f2_10 sf .232(B)A f0_10 sf .351 .035( given )J f2_10 sf .252(C)A f0_10 sf .304 .03(. A vertex Y is a )J f2_10 sf .14(collider)A f0_10 sf .261 .026( on an )J 308 393 :M -.27(undirected )A 351 393 :M .202 .02(path )J 372 393 :M .255 .026(U )J 383 393 :M .328 .033(if )J 393 393 :M .255 .026(U )J 404 393 :M .098 .01(contains )J 441 393 :M .056 .006(a )J 449 393 :M .232 .023(subpath )J 484 393 :M (X)S 59 405 :M f1_10 sf S f0_10 sf (Y )S f1_10 sf S f0_10 sf .052 .005( Z. Say that a vertex V on an undirected path )J 272 405 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 283 405 :M -.116(between )A 319 405 :M .255 .026(X )J 330 405 :M -.313(and )A 347 405 :M .255 .026(Y )J 358 405 :M .694 .069(is )J 369 405 :M f2_10 sf .684(active)A f0_10 sf .411 .041( )J 402 405 :M .417 .042(on )J 416 405 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 427 405 :M .189 .019(given )J 453 405 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 464 405 :M .134<28>A f2_10 sf .268(Z)A f0_10 sf .1 .01( )J 478 405 :M .111(not)A 59 417 :M .125 .012(containing X and Y\) if and only if either V is not a collider on )J f4_10 sf .077(U)A f0_10 sf .092 .009( and not in )J f2_10 sf .071(Z)A f0_10 sf .049 .005(, )J 378 417 :M .144 .014(or )J 390 417 :M .255 .026(V )J 401 417 :M .694 .069(is )J 412 417 :M .056 .006(a )J 420 417 :M -.13(collider )A 453 417 :M .417 .042(on )J 467 417 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 478 417 :M -.719(and)A 59 429 :M .262 .026(is an ancestor of )J f2_10 sf .149(Z)A f0_10 sf .294 .029(. For three disjoint sets of variables )J f2_10 sf .162(A)A f0_10 sf .093 .009(, )J f2_10 sf .149(B)A f0_10 sf .102 .01(, )J 307 429 :M -.313(and )A 324 429 :M f2_10 sf 1.052(C)A f0_10 sf .662 .066(, )J 339 429 :M f2_10 sf .951(A)A f0_10 sf .329 .033( )J 351 429 :M .694 .069(is )J 362 429 :M f2_10 sf .421(d-connected)A f0_10 sf .224 .022( )J 422 429 :M .601 .06(to )J 434 429 :M f2_10 sf .604(B)A f0_10 sf .226 .023( )J 445 429 :M .189 .019(given )J 471 429 :M f2_10 sf .951(C)A f0_10 sf .329 .033( )J 483 429 :M .222(in)A 59 441 :M .311 .031(graph )J f4_10 sf .142(G)A f0_10 sf .112 .011(, if )J 107 441 :M -.313(and )A 124 441 :M .515 .052(only )J 146 441 :M .328 .033(if )J 156 441 :M -.097(there )A 179 441 :M .694 .069(is )J 190 441 :M .051 .005(an )J 203 441 :M -.27(undirected )A 246 441 :M .202 .02(path )J 267 441 :M .047 .005(from )J 290 441 :M .281 .028(some )J 315 441 :M -.044(member )A 351 441 :M .144 .014(of )J 363 441 :M f2_10 sf .951(A)A f0_10 sf .329 .033( )J 375 441 :M .601 .06(to )J 387 441 :M .056 .006(a )J 395 441 :M -.044(member )A 431 441 :M .144 .014(of )J 443 441 :M f2_10 sf .604(B)A f0_10 sf .226 .023( )J 454 441 :M .123 .012(such )J 476 441 :M (that)S 59 453 :M .072 .007(every vertex on )J f4_10 sf (U)S f0_10 sf .026 .003( is )J 144 453 :M -.062(active )A 171 453 :M .189 .019(given )J 197 453 :M f2_10 sf .868(C)A f0_10 sf .577 .058(; )J 212 453 :M -.052(for )A 227 453 :M -.097(three )A 250 453 :M .276 .028(disjoint )J 284 453 :M .361 .036(sets )J 303 453 :M .144 .014(of )J 315 453 :M -.065(variables )A 354 453 :M f2_10 sf 1.052(A)A f0_10 sf .662 .066(, )J 369 453 :M f2_10 sf .76(B)A f0_10 sf .518 .052(, )J 383 453 :M -.313(and )A 400 453 :M f2_10 sf 1.052(C)A f0_10 sf .662 .066(, )J 415 453 :M f2_10 sf .951(A)A f0_10 sf .329 .033( )J 427 453 :M .694 .069(is )J 438 453 :M f2_10 sf .246(d-separated)A 59 465 :M f0_10 sf .316 .032(from )J f2_10 sf .144(B)A f0_10 sf .218 .022( given )J f2_10 sf .156(C)A f0_10 sf .206 .021( in graph )J f4_10 sf .156(G)A f0_10 sf .204 .02(, if and only )J f2_10 sf .156(A)A f0_10 sf .263 .026( is not d-connected to )J f2_10 sf .144(B)A f0_10 sf .218 .022( given )J f2_10 sf .156(C)A f0_10 sf (.)S 59 480 :M .745 .074(Two DAGs are )J f2_10 sf 1.291 .129(d-separation )J 186 480 :M .574(equivalent)A f0_10 sf .323 .032( )J 240 480 :M .328 .033(if )J 250 480 :M -.313(and )A 267 480 :M .515 .052(only )J 289 480 :M .328 .033(if )J 299 480 :M .202 .02(they )J 320 480 :M -.094(have )A 342 480 :M .218 .022(the )J 358 480 :M (same )S 382 480 :M -.01(vertices, )A 419 480 :M -.313(and )A 436 480 :M .202 .02(they )J 457 480 :M -.094(have )A 479 480 :M -.108(the)A 59 492 :M -.03(same d-separation relations. If two )A 200 492 :M (DAGs )S 229 492 :M f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 243 492 :M -.313(and )A 260 492 :M f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 274 492 :M -.235(are )A 289 492 :M -.16(d-separation )A 340 492 :M -.059(equivalent, )A 387 492 :M .202 .02(then )J 408 492 :M .202 .02(they )J 429 492 :M .18 .018(entail )J 455 492 :M .218 .022(the )J 471 492 :M -.182(same)A 59 504 :M -.082(set of conditional independence relations, and )A 242 504 :M .218 .022(the )J 258 504 :M .303 .03(set )J 273 504 :M .144 .014(of )J 285 504 :M .218 .022(distributions )J 339 504 :M .361 .036(that )J 358 504 :M .235 .023(satisfy )J 388 504 :M .218 .022(the )J 404 504 :M .044 .004(local )J 427 504 :M -.337(directed )A 460 504 :M -.131(Markov)A 59 516 :M -.046(property )A 97 516 :M -.052(for )A 113 516 :M f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 128 516 :M -.174(equals )A 157 516 :M .218 .022(the )J 174 516 :M .303 .03(set )J 190 516 :M .144 .014(of )J 203 516 :M .178 .018(distribution )J 254 516 :M .361 .036(that )J 275 516 :M .235 .023(satisfy )J 307 516 :M .218 .022(the )J 325 516 :M .044 .004(local )J 350 516 :M -.337(directed )A 385 516 :M -.026(Markov )A 422 516 :M -.046(property )A 461 516 :M -.052(for )A 478 516 :M f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf (.)S 59 528 :M -.06(Theorems )A 102 528 :M .361 .036(that )J 121 528 :M -.149(provide )A 154 528 :M .218 .022(the )J 170 528 :M .337 .034(basis )J 194 528 :M -.052(for )A 209 528 :M .474 .047(polynomial )J 259 528 :M .519 .052(time )J 282 528 :M .366 .037(algorithms )J 330 528 :M -.052(for )A 346 528 :M .493 .049(testing )J 378 528 :M -.16(d-separation )A 430 528 :M -.204(equivalence )A 480 528 :M -.328(for)A 59 540 :M -.076(DAGs was given in Verma and Pearl \(1990\), for cyclic directed graphs in Richardson \(1994\), and for directed)A 59 552 :M -.014(acyclic graphs with latent variables in Spirtes and Verma \(1992\).)A 59 567 :M (DAGs )S 88 567 :M -.235(are )A 103 567 :M .281 .028(also )J 123 567 :M -.207(used )A 144 567 :M .601 .06(to )J 156 567 :M -.126(represent )A 195 567 :M -.081(causal )A 223 567 :M -.031(processes. )A 267 567 :M -.297(Under )A 294 567 :M .753 .075(this )J 313 567 :M .055 .005(interpretation, )J 373 567 :M .056 .006(a )J 381 567 :M -.337(directed )A 414 567 :M -.344(edge )A 436 567 :M .047 .005(from )J 460 567 :M .255 .026(A )J 472 567 :M .601 .06(to )J 485 567 :M (B)S 59 579 :M (means )S 89 579 :M .361 .036(that )J 109 579 :M .255 .026(A )J 121 579 :M .694 .069(is )J 133 579 :M .056 .006(a )J 142 579 :M -.21(direct )A 168 579 :M -.141(cause )A 194 579 :M .144 .014(of )J 207 579 :M -.17(B )A 218 579 :M -.06(relative )A 252 579 :M .601 .06(to )J 265 579 :M .218 .022(the )J 282 579 :M -.065(variables )A 322 579 :M .601 .06(in )J 335 579 :M .218 .022(the )J 352 579 :M .244 .024(DAG. )J 382 579 :M .361 .036(Suppose )J 422 579 :M .056 .006(a )J 432 579 :M -.081(causal )A 462 579 :M -.164(process)A 59 591 :M -.234(represented )A 106 591 :M .417 .042(by )J 120 591 :M -.053(DAG )A 145 591 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 156 591 :M -.139(generates )A 196 591 :M .281 .028(some )J 221 591 :M .363 .036(population )J 268 591 :M .517 .052(with )J 290 591 :M .056 .006(a )J 298 591 :M .189 .019(given )J 324 591 :M .178 .018(distribution )J 374 591 :M f4_10 sf .14(P)A f0_10 sf .077<28>A f2_10 sf .166(V)A f0_10 sf .122 .012(\) )J 398 591 :M .361 .036(that )J 418 591 :M .154 .015(satisfies )J 455 591 :M .218 .022(the )J 472 591 :M -.108(local)A 59 603 :M -.337(directed )A 93 603 :M -.026(Markov )A 129 603 :M -.046(property )A 167 603 :M -.052(for )A 183 603 :M f4_10 sf .461(G)A f0_10 sf .29 .029(. )J 198 603 :M -.078(If )A 209 603 :M .281 .028(some )J 235 603 :M .144 .014(of )J 248 603 :M .218 .022(the )J 265 603 :M -.065(variables )A 305 603 :M .601 .06(in )J 318 603 :M f2_10 sf .951(V)A f0_10 sf .329 .033( )J 331 603 :M -.235(are )A 348 603 :M -.119(unmeasured, )A 403 603 :M -.313(and )A 422 603 :M .281 .028(some )J 449 603 :M -.094(have )A 473 603 :M -.292(been)A 59 615 :M -.156(conditioned )A 108 615 :M .417 .042(on )J 122 615 :M -.316(\(due )A 142 615 :M .601 .06(to )J 154 615 :M .263 .026(those )J 179 615 :M -.065(variables )A 218 615 :M .189 .019(being )J 244 615 :M -.033(causally )A 280 615 :M -.243(related )A 309 615 :M .601 .06(to )J 321 615 :M .218 .022(the )J 337 615 :M .465 .047(sampling )J 378 615 :M -.037(mechanism\) )A 430 615 :M .202 .02(then )J 451 615 :M .218 .022(the )J 467 615 :M .303 .03(set )J 483 615 :M -.328(of)A 59 627 :M -.114(conditional independence relations )A 198 627 :M -.144(entailed )A 232 627 :M -.052(for )A 247 627 :M .218 .022(the )J 263 627 :M .315 .032(subset )J 292 627 :M .144 .014(of )J 304 627 :M -.226(measured )A 344 627 :M -.065(variables )A 383 627 :M .601 .06(in )J 395 627 :M .218 .022(the )J 411 627 :M .364 .036(subpopulation )J 472 627 :M -.145(from)A 59 639 :M .041 .004(which the sample is drawn is not necessarily )J 241 639 :M -.231(equal )A 265 639 :M .601 .06(to )J 277 639 :M .218 .022(the )J 293 639 :M .303 .03(set )J 308 639 :M .144 .014(of )J 320 639 :M -.045(conditional )A 368 639 :M -.289(independence )A 423 639 :M .038 .004(relations )J 461 639 :M -.236(entailed)A 59 651 :M .417 .042(by )J 73 651 :M .047 .005(any )J 91 651 :M -.053(DAG )A 116 651 :M .343 .034(\(without )J 155 651 :M .18 .018(latent )J 182 651 :M -.065(variables )A 222 651 :M .144 .014(or )J 235 651 :M (selection )S 275 651 :M .354 .035(bias\). )J 302 651 :M .178 .018(Assume )J 339 651 :M .202 .02(then )J 361 651 :M .361 .036(that )J 381 651 :M .218 .022(the )J 398 651 :M -.065(variables )A 438 651 :M .601 .06(in )J 451 651 :M f2_10 sf .951(V)A f0_10 sf .329 .033( )J 464 651 :M -.126(can )A 482 651 :M -.439(be)A 59 663 :M -.074(partitioned )A 106 663 :M .674 .067(into )J 127 663 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 140 663 :M -.159(\(observed\), )A 188 663 :M f2_10 sf .604(L)A f0_10 sf .226 .023( )J 200 663 :M .07 .007(\(latent\), )J 236 663 :M -.313(and )A 254 663 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 266 663 :M -.153(\(selected, )A 307 663 :M .144 .014(or )J 320 663 :M -.156(conditioned )A 371 663 :M .48 .048(on.\) )J 393 663 :M .144 .014(In )J 407 663 :M .361 .036(that )J 428 663 :M -.176(case )A 450 663 :M -.118(instead )A 483 663 :M -.328(of)A 59 675 :M .035 .003(observing )J 102 675 :M f4_10 sf .249(P)A f0_10 sf .135<28>A f2_10 sf .294(V)A f0_10 sf .282 .028(\), )J 129 675 :M -.079(we )A 144 675 :M .218 .022(may )J 165 675 :M .051 .005(be )J 178 675 :M -.039(able )A 198 675 :M .601 .06(to )J 210 675 :M -.085(observe )A 244 675 :M .515 .052(only )J 266 675 :M f4_10 sf .387(P)A f0_10 sf .211<28>A f2_10 sf .493(O)A f0_10 sf .127(|)A f2_10 sf .352(S)A f0_10 sf .158 .016( )J 296 675 :M -.14(= )A 305 675 :M f2_10 sf .627(1)A f0_10 sf .871 .087(\), )J 321 675 :M .361 .036(that )J 340 675 :M .694 .069(is )J 351 675 :M .218 .022(the )J 367 675 :M (marginal )S 406 675 :M .178 .018(distribution )J 457 675 :M -.067(over )A 479 675 :M -.108(the)A 59 687 :M -.011(observed variables in the selected subpopulation. Let us call )A 302 687 :M f4_10 sf .387(P)A f0_10 sf .211<28>A f2_10 sf .493(O)A f0_10 sf .127(|)A f2_10 sf .352(S)A f0_10 sf .158 .016( )J 332 687 :M -.14(= )A 341 687 :M f2_10 sf .541(1)A f0_10 sf .574 .057(\) )J 354 687 :M .218 .022(the )J 370 687 :M -.247(\322observed\323 )A 416 687 :M .387 .039(distribution. )J 469 687 :M -.328(There)A endp %%Page: 2 2 %%BeginPageSetup initializepage (peter; page: 2 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (2)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf -.235(are )A 76 51 :M .366 .037(algorithms )J 125 51 :M .352 .035(which, )J 158 51 :M -.253(under )A 185 51 :M .281 .028(some )J 212 51 :M .21 .021(plausible )J 254 51 :M .504 .05(assumptions )J 310 51 :M (relating )S 346 51 :M .295 .03(probability )J 396 51 :M .218 .022(distributions )J 452 51 :M .601 .06(to )J 467 51 :M -.197(causal)A 59 63 :M -.011(processes, are correct in the large sample limit, and that can construct a representation of the )A 431 63 :M .044 .004(class )J 454 63 :M .144 .014(of )J 466 63 :M -.182(DAGs)A 59 75 :M .118 .012(\(that )J 82 75 :M .218 .022(may )J 104 75 :M -.094(have )A 128 75 :M .18 .018(latent )J 156 75 :M -.065(variables )A 197 75 :M -.313(and )A 216 75 :M -.065(variables )A 257 75 :M -.156(conditioned )A 308 75 :M .133 .013(on\) )J 327 75 :M .361 .036(that )J 348 75 :M -.235(are )A 365 75 :M .036 .004(compatible )J 415 75 :M .517 .052(with )J 439 75 :M .218 .022(the )J 457 75 :M -.299(observed)A 59 87 :M -.045(conditional )A 107 87 :M -.289(independence )A 162 87 :M .286 .029(relations. )J 203 87 :M .049 .005(See )J 221 87 :M .429 .043(Spirtes )J 253 87 :M .236 .024(et )J 264 87 :M .602 .06(al. )J 278 87 :M .357 .036(1993 )J 303 87 :M -.052(for )A 319 87 :M .218 .022(the )J 336 87 :M .18 .018(latent )J 363 87 :M -.087(variable )A 399 87 :M -.176(case )A 420 87 :M .556 .056(without )J 456 87 :M -.067(selection)A 59 99 :M .099(bias.)A 59 114 :M .155 .016(For a given DAG )J f4_10 sf .087(G)A f0_10 sf .142 .014(, and a partition of the variable set )J f2_10 sf .087(V)A f0_10 sf ( )S 291 114 :M .144 .014(of )J 303 114 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 314 114 :M .674 .067(into )J 334 114 :M -.199(observed )A 372 114 :M .097<28>A f2_10 sf .227(O)A f0_10 sf .202 .02(\), )J 393 114 :M (selection )S 432 114 :M .345<28>A f2_10 sf .577(S)A f0_10 sf .72 .072(\), )J 452 114 :M -.313(and )A 469 114 :M -.042(latent)A 59 126 :M .082<28>A f2_10 sf .165(L)A f0_10 sf .328 .033(\) variables, we will write )J f4_10 sf .178(G)A f0_10 sf .082<28>A f2_10 sf .192(O)A f0_10 sf .062(,)A f2_10 sf .137(S)A f0_10 sf .062(,)A f2_10 sf .165(L)A f0_10 sf .258 .026(\). Let us now )J 268 126 :M -.193(extend )A 297 126 :M .218 .022(the )J 313 126 :M -.038(definition )A 355 126 :M .144 .014(of )J 367 126 :M -.16(d-separation )A 418 126 :M -.204(equivalence )A 467 126 :M .601 .06(to )J 479 126 :M -.108(the)A 59 138 :M -.176(case )A 81 138 :M -.185(where )A 110 138 :M -.097(there )A 135 138 :M .218 .022(may )J 158 138 :M .051 .005(be )J 174 138 :M .18 .018(latent )J 203 138 :M -.065(variables )A 245 138 :M -.313(and )A 265 138 :M (selection )S 307 138 :M .596 .06(bias. )J 333 138 :M .132 .013(Two )J 358 138 :M -.337(directed )A 394 138 :M -.026(graphs )A 427 138 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .313 .031(\) )J 478 138 :M -.719(and)A 59 150 :M f4_10 sf .379(G)A f0_6 sf 0 2 rm .157(2)A 0 -2 rm f0_10 sf .175<28>A f2_10 sf .291<4FD5>A f0_10 sf .131(,)A f2_10 sf .262<4CD5>A f0_10 sf .131(,)A f2_10 sf .233<53D5>A f0_10 sf .414 .041(\) are )J f2_10 sf 1.603 .16(d-separation equivalent)J f0_10 sf .453 .045( if and only if )J f2_10 sf .408(O)A f0_10 sf .243 .024( = )J f2_10 sf .291<4FD5>A f0_10 sf .472 .047(, and for all )J f2_10 sf .379(X)A f0_10 sf .219 .022(, )J f2_10 sf .379(Y)A f0_10 sf .408 .041( and )J f2_10 sf .35(Z)A f0_10 sf .607 .061( included in )J f2_10 sf .408(O)A f0_10 sf (,)S 59 162 :M f2_10 sf .122(X)A f0_10 sf .131 .013( and )J f2_10 sf .122(Y)A f0_10 sf .242 .024( are d-separated given )J f2_10 sf .113(Z)A f0_10 sf ( )S f1_10 sf .13A f0_10 sf ( )S f2_10 sf .094(S)A f0_10 sf .09 .009( in )J f4_10 sf .122(G)A f0_6 sf 0 2 rm .051(1)A 0 -2 rm f0_10 sf .056<28>A f2_10 sf .131(O)A f0_10 sf (,)S f2_10 sf .113(L)A f0_10 sf (,)S f2_10 sf .094(S)A f0_10 sf .152 .015(\) if and only if )J f2_10 sf .122(X)A f0_10 sf .131 .013( and )J f2_10 sf .122(Y)A f0_10 sf .242 .024( are d-separated given )J f2_10 sf .113(Z)A f0_10 sf ( )S f1_10 sf .13A f0_10 sf ( )S f2_10 sf .075<53D5>A f0_10 sf ( )S 483 162 :M .222(in)A 59 174 :M f4_10 sf .652(G)A f0_6 sf 0 2 rm .271(2)A 0 -2 rm f0_10 sf .3<28>A f2_10 sf .501<4FD5>A f0_10 sf .226(,)A f2_10 sf .451<4CD5>A f0_10 sf .226(,)A f2_10 sf .401<53D5>A f0_10 sf .627 .063(\). )J 122 174 :M .61 .061(Intuitively, )J 171 174 :M .218 .022(the )J 187 174 :M -.045(conditional )A 235 174 :M -.289(independence )A 291 174 :M .038 .004(relations )J 330 174 :M (true )S 350 174 :M .601 .06(in )J 363 174 :M .218 .022(the )J 380 174 :M -.199(observed )A 419 174 :M .178 .018(distribution )J 470 174 :M -.304(could)A 59 186 :M -.077(have been generated either by the causal )A 221 186 :M -.053(DAG )A 246 186 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .313 .031(\) )J 294 186 :M .144 .014(or )J 306 186 :M .417 .042(by )J 320 186 :M f4_10 sf .652(G)A f0_6 sf 0 2 rm .271(2)A 0 -2 rm f0_10 sf .3<28>A f2_10 sf .501<4FD5>A f0_10 sf .226(,)A f2_10 sf .451<4CD5>A f0_10 sf .226(,)A f2_10 sf .401<53D5>A f0_10 sf .627 .063(\). )J 383 186 :M -.023(Information )A 434 186 :M .753 .075(just )J 453 186 :M .189 .019(about )J 479 186 :M -.108(the)A 59 198 :M -.199(observed )A 98 198 :M -.045(conditional )A 147 198 :M -.289(independence )A 203 198 :M .038 .004(relations )J 243 198 :M -.026(cannot )A 275 198 :M .289 .029(distinguish )J 325 198 :M .047 .005(any )J 345 198 :M .387 .039(two )J 366 198 :M (DAGs )S 397 198 :M .043 .004(which )J 427 198 :M -.235(are )A 444 198 :M -.22(d-separation)A 59 210 :M -.115(equivalent.)A 59 225 :M -.085(In order to state necessary and sufficient conditions for d-separation equivalence, )A 380 225 :M -.079(we )A 395 225 :M .676 .068(will )J 415 225 :M -.344(need )A 436 225 :M .218 .022(the )J 452 225 :M .015(following)A 59 237 :M -.021(concept. A mixed ancestral graph \(MAG\) is an extended graph consisting of a set of vertices )A f2_10 sf (V)S f0_10 sf -.022(, and )A 460 237 :M .056 .006(a )J 468 237 :M .303 .03(set )J 483 237 :M -.328(of)A 59 249 :M -.036(edges between vertices, where there may be the following kinds of )A 327 249 :M -.174(edges: )A 355 249 :M .255 .026(A )J 366 249 :M f1_10 sf .065A f0_10 sf ( )S 380 249 :M .275 .028(B, )J 393 249 :M .255 .026(A )J 404 249 :M .111(o)A f1_10 sf .222A f0_10 sf .151 .015(o )J 428 249 :M .275 .028(B, )J 441 249 :M .255 .026(A )J 452 249 :M .182(o)A f1_10 sf .359A f0_10 sf .091 .009( )J 471 249 :M .275 .028(B, )J 484 249 :M (A)S 59 261 :M f1_10 sf .346A f0_10 sf .274 .027(o B, A )J f1_10 sf .346A f0_10 sf .216 .022( B, )J 125 261 :M .144 .014(or )J 137 261 :M .255 .026(A )J 148 261 :M f1_10 sf .504A f0_10 sf .128 .013( )J 162 261 :M .275 .028(B. )J 175 261 :M (\(A )S 189 261 :M .132 .013(MAG )J 216 261 :M .218 .022(may )J 237 261 :M .051 .005(be )J 250 261 :M -.281(considered )A 294 261 :M .056 .006(a )J 302 261 :M -.037(special )A 333 261 :M -.176(case )A 353 261 :M .144 .014(of )J 365 261 :M .056 .006(a )J 373 261 :M .387 .039(PAG )J 397 261 :M .361 .036(that )J 416 261 :M -.103(represents )A 459 261 :M .056 .006(a )J 467 261 :M .023(single)A 59 273 :M .035 .003(graph. See Richardson 1996.\) We say that the A endpoint of an A )J f1_10 sf S f0_10 sf .03 .003( B edge is \322\320\323; the A endpoint of an A)J 59 285 :M f1_10 sf .107A f0_10 sf .068 .007( B, A )J f1_10 sf .101A f0_10 sf .085 .008(o B, or. A )J f1_10 sf .101A f0_10 sf .102 .01( B edge is \322<\323; and )J 237 285 :M .218 .022(the )J 253 285 :M .255 .026(A )J 264 285 :M -.062(endpoint )A 302 285 :M .144 .014(of )J 314 285 :M .051 .005(an )J 327 285 :M .255 .026(A )J 338 285 :M .111(o)A f1_10 sf .222A f0_10 sf .151 .015(o )J 362 285 :M -.17(B )A 372 285 :M .144 .014(or )J 384 285 :M .255 .026(A )J 395 285 :M .182(o)A f1_10 sf .359A f0_10 sf .091 .009( )J 414 285 :M -.17(B )A 424 285 :M -.344(edge )A 445 285 :M .694 .069(is )J 456 285 :M .088 .009J 476 285 :M -.273(The)A 59 297 :M -.029(conventions for the B endpoints are analogous. A mixed ancestral graph )A 349 297 :M -.052(for )A 364 297 :M .056 .006(a )J 372 297 :M .303 .03(set )J 387 297 :M .144 .014(of )J 399 297 :M -.026(graphs )A 429 297 :M f2_10 sf .546(G)A f0_10 sf .175 .018( )J 441 297 :M -.204(each )A 462 297 :M -.072(sharing)A 59 309 :M .15 .015(the same set )J 112 309 :M .144 .014(of )J 124 309 :M -.199(observed )A 162 309 :M -.087(variable )A 197 309 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 209 309 :M .098 .01(contains )J 246 309 :M .139 .014(information )J 297 309 :M .189 .019(about )J 323 309 :M .218 .022(the )J 339 309 :M -.101(ancestor )A 375 309 :M .038 .004(relations )J 413 309 :M .601 .06(in )J 425 309 :M f2_10 sf .744(G)A f0_10 sf .435 .043(, )J 440 309 :M .041 .004(namely )J 473 309 :M .074(only)A 59 321 :M .263 .026(those )J 85 321 :M -.101(ancestor )A 122 321 :M .038 .004(relations )J 161 321 :M .315 .032(common )J 201 321 :M .601 .06(to )J 214 321 :M .388 .039(all )J 229 321 :M -.021(members )A 270 321 :M .144 .014(of )J 283 321 :M f2_10 sf .744(G)A f0_10 sf .435 .043(. )J 299 321 :M .144 .014(In )J 312 321 :M .218 .022(the )J 329 321 :M .325 .033(following )J 373 321 :M .057 .006(definition, )J 419 321 :M .043 .004(which )J 448 321 :M -.117(provides )A 487 321 :M (a)S 59 333 :M .009 .001(semantics for MAGs we use \322*\323 as a meta-symbol indicating the )J 322 333 :M -.184(presence )A 359 333 :M .144 .014(of )J 371 333 :M .047 .005(any )J 389 333 :M .047 .005(one )J 407 333 :M .144 .014(of )J 419 333 :M 2 .2({o,\312\320, )J 448 333 :M .431 .043(>}, )J 465 333 :M .758 .076(e.g. )J 484 333 :M (A)S 59 345 :M (*)S f1_10 sf S f0_10 sf .056 .006( B represents either A )J f1_10 sf S f0_10 sf .032 .003( B, or A )J f1_10 sf S f0_10 sf .044 .004( B.)J 59 360 :M f2_10 sf 2.634 .263(Mixed Ancestral Graphs)J f0_10 sf .509 .051( \()J f2_10 sf .742(MAGs)A f0_10 sf <29>S 59 375 :M -.005(A MAG represents directed acyclic graph )A f4_10 sf (G)S f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf -.005(\) \(in which case we write MAG\()A f4_10 sf (G)S f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf (\)\) if:)S 77 390 :M .051 .005(\(i\) )J 90 390 :M -.078(If )A 100 390 :M .255 .026(A )J 111 390 :M -.313(and )A 128 390 :M -.17(B )A 138 390 :M -.235(are )A 153 390 :M .601 .06(in )J 165 390 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 180 390 :M -.097(there )A 203 390 :M .694 .069(is )J 214 390 :M .051 .005(an )J 227 390 :M -.344(edge )A 248 390 :M -.116(between )A 284 390 :M .255 .026(A )J 295 390 :M -.313(and )A 312 390 :M -.17(B )A 322 390 :M .601 .06(in )J 334 390 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 409 390 :M .328 .033(if )J 420 390 :M -.313(and )A 438 390 :M .515 .052(only )J 461 390 :M -.052(for )A 477 390 :M -.219(any)A 77 402 :M .178 .018(subset )J f2_10 sf .11(W)A f0_10 sf ( )S f1_10 sf .078A f0_10 sf ( )S f2_10 sf .085(O)A f0_10 sf .163 .016(\\{A,B}, A and B are d-connected given )J f2_10 sf .11(W)A f0_10 sf .025 .002( )J f1_10 sf .084A f0_10 sf ( )S f2_10 sf .061(S)A f0_10 sf .058 .006( in )J f4_10 sf .079(G)A f0_10 sf <28>S f2_10 sf .085(O)A f0_10 sf (,)S f2_10 sf .061(S)A f0_10 sf (,)S f2_10 sf .073(L)A f0_10 sf .064(\).)A 77 417 :M .137 .014(\(ii\) There is an edge)J f3_10 sf ( )S f0_10 sf .084 .008(A )J f1_10 sf .103A f0_10 sf .069 .007( B or B )J f1_10 sf .103A f0_10 sf .128 .013( A in MAG\()J f4_10 sf .075(G)A f0_10 sf <28>S f2_10 sf .081(O)A f0_10 sf (,)S f2_10 sf .058(S)A f0_10 sf (,)S f2_10 sf .069(L)A f0_10 sf .035<2929>A f3_10 sf ( )S f0_10 sf .108 .011(if and only if A is an ancestor of B )J 461 417 :M .555 .056(but )J 478 417 :M .111(not)A 77 429 :M f2_10 sf .463(S)A f0_10 sf .444 .044( in )J f4_10 sf .602(G)A f0_10 sf .277<28>A f2_10 sf .649(O)A f0_10 sf .208(,)A f2_10 sf .463(S)A f0_10 sf .208(,)A f2_10 sf .556(L)A f0_10 sf .509(\);)A 77 444 :M .234 .023(\(iii\) There is an edge)J f3_10 sf ( )S f0_10 sf .141 .014(A )J f1_10 sf .172A f0_10 sf .145 .015(* B or B *)J f1_10 sf .172A f0_10 sf .214 .021( A in MAG\()J f4_10 sf .126(G)A f0_10 sf .058<28>A f2_10 sf .136(O)A f0_10 sf (,)S f2_10 sf .097(S)A f0_10 sf (,)S f2_10 sf .116(L)A f0_10 sf .058<2929>A f3_10 sf ( )S f0_10 sf .16 .016(if and only if A is )J f2_10 sf .081(not)A f0_10 sf .209 .021( an ancestor )J 473 444 :M .144 .014(of )J 485 444 :M (B)S 77 456 :M .608 .061(or )J f2_10 sf .406(S)A f0_10 sf .389 .039( in )J f4_10 sf .527(G)A f0_10 sf .243<28>A f2_10 sf .568(O)A f0_10 sf .183(,)A f2_10 sf .406(S)A f0_10 sf .183(,)A f2_10 sf .487(L)A f0_10 sf .446(\);)A 77 471 :M .192 .019(\(iv\) There is an edge)J f3_10 sf ( )S f0_10 sf .161 .016(A o)J f1_10 sf .142A f0_10 sf .118 .012(* B or B *)J f1_10 sf .142A f0_10 sf .187 .019(o A in MAG\()J f4_10 sf .102(G)A f0_10 sf <28>S f2_10 sf .11(O)A f0_10 sf (,)S f2_10 sf .079(S)A f0_10 sf (,)S f2_10 sf .095(L)A f0_10 sf .047<2929>A f3_10 sf ( )S f0_10 sf .15 .015(if and only if A is an ancestor of )J f2_10 sf .079(S)A f0_10 sf .121 .012( in)J 77 483 :M f4_10 sf .457(G)A f0_10 sf .211<28>A f2_10 sf .492(O)A f0_10 sf .158(,)A f2_10 sf .352(S)A f0_10 sf .158(,)A f2_10 sf .422(L)A f0_10 sf .369(\).)A 59 498 :M -.043(The definition of \322d-separation\323 given for DAGs can be applied directly to MAGs, as )A 401 498 :M .515 .052(long )J 423 498 :M .144 .014(as )J 435 498 :M .123 .012(such )J 457 498 :M -.14(concepts)A 59 510 :M -.024(as \322undirected path\323, \322collider\323, etc., are given their obvious extensions )A 350 510 :M .601 .06(to )J 362 510 :M .189 .019(apply )J 388 510 :M .601 .06(to )J 400 510 :M .522 .052(MAGs. )J 434 510 :M -.188(We )A 451 510 :M -.133(include )A 483 510 :M .222(in)A 59 522 :M -.044(the Appendix the definitions of terms such as \322undirected path\323 etc. which apply both to directed graphs and)A 59 534 :M .071(MAGs.)A 59 549 :M (The )S 78 549 :M .266 .027(first )J 98 549 :M .281 .028(step )J 118 549 :M .601 .06(in )J 130 549 :M .169 .017(forming )J 166 549 :M .056 .006(a )J 174 549 :M .132 .013(MAG )J 201 549 :M -.052(for )A 216 549 :M .056 .006(a )J 224 549 :M -.053(graph )A 250 549 :M .694 .069(is )J 261 549 :M .601 .06(to )J 273 549 :M .047 .005(form )J 296 549 :M .218 .022(the )J 312 549 :M -.101(ancestor )A 348 549 :M .249 .025(matrix )J 378 549 :M -.052(for )A 393 549 :M .218 .022(the )J 409 549 :M .146 .015(graph. )J 438 549 :M .134 .013(Let )J 455 549 :M .454 .045(n )J 465 549 :M .051 .005(be )J 479 549 :M -.108(the)A 59 561 :M .357 .036(number of vertices in )J f2_10 sf .196(O)A f0_10 sf .057 .006( )J f1_10 sf .194A f0_10 sf .057 .006( )J f2_10 sf .14(S)A f0_10 sf .057 .006( )J f1_10 sf .194A f0_10 sf .057 .006( )J f2_10 sf .168(L)A f0_10 sf .279 .028( and m the number of )J 287 561 :M -.074(vertices )A 321 561 :M .601 .06(in )J 333 561 :M f2_10 sf .744(O)A f0_10 sf .435 .043(. )J 348 561 :M .558 .056(Aho, )J 372 561 :M -.01(Hopcroft, )A 414 561 :M -.313(and )A 431 561 :M .317 .032(Ullman )J 465 561 :M -.131(\(1974\))A 59 573 :M .086 .009(describes a transitive closure algorithm for filling in such a matrix that )J 346 573 :M .694 .069(is )J 357 573 :M .024(O\(n)A f0_6 sf 0 -4 rm (3)S 0 4 rm f0_10 sf (\). )S 385 573 :M (Then )S 409 573 :M -.204(each )A 430 573 :M (pair )S 449 573 :M .144 .014(of )J 461 573 :M -.156(vertices)A 59 585 :M .041 .004(X and Y in )J f2_10 sf (O)S f0_10 sf .071 .007( \(O\(m)J f0_6 sf 0 -4 rm (2)S 0 4 rm f0_10 sf .065 .007(\)\) is adjacent in MAG\()J f4_10 sf (G)S f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf .039 .004(\)\) if and )J 307 585 :M .515 .052(only )J 329 585 :M .328 .033(if )J 339 585 :M .202 .02(they )J 360 585 :M -.235(are )A 375 585 :M .555 .056(not )J 392 585 :M -.325(d-separated )A 438 585 :M -.157(\(O\(n)A f0_6 sf 0 -4 rm -.1(2)A 0 4 rm f0_10 sf -.152(\)\) )A 469 585 :M -.054(given)A 59 597 :M .175<28>A f2_10 sf .247(Ancestors)A f0_10 sf 1.066 .107(\({X,Y} )J 142 597 :M f1_10 sf .62A f0_10 sf .202 .02( )J 154 597 :M f2_10 sf .788(S)A f0_10 sf .751 .075(\) )J 168 597 :M f1_10 sf .62A f0_10 sf .202 .02( )J 180 597 :M f2_10 sf .225(O)A f0_10 sf .153 .015(\) )J 195 597 :M .601 .06(in )J 207 597 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .313 .031(\) )J 255 597 :M -.209(\(where )A 285 597 :M f2_10 sf .483(Ancestors)A f0_10 sf .343<28>A f2_10 sf .687(Z)A f0_10 sf .546 .055(\) )J 351 597 :M .694 .069(is )J 363 597 :M .218 .022(the )J 380 597 :M .303 .03(set )J 396 597 :M .144 .014(of )J 409 597 :M -.074(vertices )A 444 597 :M .043 .004(which )J 473 597 :M -.292(have)A 59 609 :M -.029(descendants in Z; see Lemma 5.\) The orientation of each )A 288 609 :M -.344(edge )A 309 609 :M .601 .06(in )J 321 609 :M .218 .022(the )J 337 609 :M .132 .013(MAG )J 364 609 :M -.129(\(O\(m)A f0_6 sf 0 -4 rm -.072(2)A 0 4 rm f0_10 sf -.109(\)\) )A 398 609 :M .202 .02(then )J 419 609 :M -.126(can )A 436 609 :M .051 .005(be )J 449 609 :M -.331(determined)A 59 621 :M .197 .02(by examining the ancestor matrix. So forming a MAG is O\(n)J f0_6 sf 0 -4 rm (3)S 0 4 rm f0_10 sf .086(m\).)A 59 636 :M -.026(If )A f4_10 sf -.062(U)A f0_10 sf -.035( is an acyclic undirected path containing X )A 248 636 :M -.313(and )A 265 636 :M .275 .028(B, )J 278 636 :M -.313(and )A 295 636 :M .255 .026(X )J 306 636 :M .694 .069(is )J 317 636 :M -.172(before )A 345 636 :M -.17(B )A 355 636 :M .417 .042(on )J 369 636 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 383 636 :M .202 .02(then )J 404 636 :M f4_10 sf -.168(U)A f0_10 sf -.119(\(X,B\) )A 437 636 :M -.103(represents )A 480 636 :M -.108(the)A 59 648 :M -.119(unique )A 90 648 :M .232 .023(subpath )J 126 648 :M .144 .014(of )J 139 648 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 151 648 :M -.116(between )A 188 648 :M .255 .026(X )J 200 648 :M -.313(and )A 218 648 :M .275 .028(B. )J 232 648 :M -.078(If )A 243 648 :M -.17(B )A 254 648 :M .694 .069(is )J 266 648 :M -.172(before )A 295 648 :M .255 .026(X )J 308 648 :M .417 .042(on )J 324 648 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 340 648 :M .417 .042(by )J 356 648 :M -.038(definition )A 400 648 :M f4_10 sf -.168(U)A f0_10 sf -.119(\(X,B\) )A 435 648 :M -.14(= )A 446 648 :M f4_10 sf -.054(U)A f0_10 sf -.035(\(B,X\). )A 484 648 :M -.328(In)A 59 660 :M .232(MAG\()A f4_10 sf .252(G)A f0_10 sf .116<28>A f2_10 sf .271(O)A f0_10 sf .087(,)A f2_10 sf .194(S)A f0_10 sf .087(,)A f2_10 sf .233(L)A f0_10 sf .29 .029(\)\), )J f4_10 sf .252(U)A f0_10 sf .18 .018( is a )J f2_10 sf 1.012 .101(discriminating path)J f0_10 sf .304 .03( for B if and only if )J f4_10 sf .252(U)A f0_10 sf .457 .046( is an undirected path between X )J 479 660 :M -.719(and)A 59 672 :M .266 .027(Y with at least )J 122 672 :M -.097(three )A 145 672 :M -.128(edges, )A 173 672 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 184 672 :M .098 .01(contains )J 221 672 :M .275 .028(B, )J 234 672 :M -.17(B )A 244 672 :M f1_10 sf .695A f0_10 sf .317 .032( )J 254 672 :M .65 .065(X, )J 268 672 :M -.17(B )A 278 672 :M -.226(adjacent )A 313 672 :M .601 .06(to )J 325 672 :M .255 .026(Y )J 336 672 :M .417 .042(on )J 350 672 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 364 672 :M .255 .026(X )J 375 672 :M .694 .069(is )J 386 672 :M .555 .056(not )J 403 672 :M -.226(adjacent )A 438 672 :M .601 .06(to )J 450 672 :M .65 .065(Y, )J 464 672 :M -.313(and )A 481 672 :M -.328(for)A 59 684 :M -.023(every vertex Q on )A f4_10 sf (U)S f0_10 sf -.022(\(X,B\) except for the endpoints Q is a collider on )A f4_10 sf (U)S f0_10 sf -.022(\(X,B\) and there is an edge Q )A f1_10 sf -.055A f0_10 sf -.034( Y )A 484 684 :M .222(in)A endp %%Page: 3 3 %%BeginPageSetup initializepage (peter; page: 3 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (3)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf .018(MAG\()A f4_10 sf (G)S f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf .027 .003(\)\). If Y is adjacent to X and Z on a path )J f4_10 sf (U)S f0_10 sf .033 .003(, and X and Z are not adjacent in thegraph, )J 464 51 :M .202 .02(then )J 485 51 :M (Y)S 59 63 :M .758 .076(is )J f2_10 sf .496(unshielded)A f0_10 sf (.)S 59 78 :M .051(MAG\()A f4_10 sf .055(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .059(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .051(L)A f0_10 sf .076 .008(\)\) and )J 152 78 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 237 78 :M -.094(have )A 259 78 :M .218 .022(the )J 275 78 :M f2_10 sf 2.746 .275(same )J 304 78 :M 3.223 .322(basic )J 334 78 :M .665(colliders)A f0_10 sf .415 .041( )J 380 78 :M .328 .033(if )J 390 78 :M -.313(and )A 407 78 :M .515 .052(only )J 429 78 :M .328 .033(if )J 439 78 :M .202 .02(they )J 460 78 :M -.094(have )A 482 78 :M -.217(\(i\))A 59 90 :M .218 .022(the )J 76 90 :M (same )S 101 90 :M -.196(adjacencies; )A 152 90 :M .206 .021(\(ii\) )J 169 90 :M .218 .022(the )J 186 90 :M (same )S 212 90 :M -.182(unshielded )A 259 90 :M -.103(colliders )A 298 90 :M .34 .034(\(iii\) )J 319 90 :M .328 .033(if )J 331 90 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 344 90 :M .694 .069(is )J 357 90 :M .056 .006(a )J 367 90 :M -.019(discriminating )A 430 90 :M .202 .02(path )J 453 90 :M -.052(for )A 470 90 :M .255 .026(X )J 483 90 :M .222(in)A 59 102 :M .185(MAG\()A f4_10 sf .201(G)A f0_6 sf 0 2 rm .083(1)A 0 -2 rm f0_10 sf .092<28>A f2_10 sf .216(O)A f0_10 sf .069(,)A f2_10 sf .154(S)A f0_10 sf .069(,)A f2_10 sf .185(L)A f0_10 sf .249 .025(\)\), )J 139 102 :M -.313(and )A 156 102 :M .218 .022(the )J 172 102 :M -.131(corresponding )A 231 102 :M .202 .02(path )J 252 102 :M f4_10 sf <55D5>S f0_10 sf ( )S 266 102 :M .601 .06(in )J 278 102 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 363 102 :M .694 .069(is )J 374 102 :M .056 .006(a )J 382 102 :M -.019(discriminating )A 443 102 :M .202 .02(path )J 465 102 :M -.052(for )A 481 102 :M .281(X,)A 59 114 :M .219 .022(then X is a collider on )J f4_10 sf .146(U)A f0_10 sf .285 .029( in MAG\()J f4_10 sf .146(G)A f0_6 sf 0 2 rm .061(1)A 0 -2 rm f0_10 sf .067<28>A f2_10 sf .158(O)A f0_10 sf .051(,)A f2_10 sf .113(S)A f0_10 sf .051(,)A f2_10 sf .135(L)A f0_10 sf .199 .02(\)\) if and only if X is a collider on )J f4_10 sf .107<55D5>A f0_10 sf .285 .029( in MAG\()J f4_10 sf .146(G)A f0_6 sf 0 2 rm .061(2)A 0 -2 rm f0_10 sf .067<28>A f2_10 sf .158(O)A f0_10 sf .051(,)A f2_10 sf .09<53D5>A f0_10 sf .051(,)A f2_10 sf .101<4CD5>A f0_10 sf .093(\)\).)A 59 129 :M f2_10 sf 2.722 .272(Theorem )J 110 129 :M .835(1:)A f0_10 sf .502 .05( )J 128 129 :M (DAGs )S 161 129 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .313 .031(\) )J 213 129 :M -.313(and )A 235 129 :M f4_10 sf .561(G)A f0_6 sf 0 2 rm .233(2)A 0 -2 rm f0_10 sf .258<28>A f2_10 sf .604(O)A f0_10 sf .194(,)A f2_10 sf .345<53D5>A f0_10 sf .194(,)A f2_10 sf .388<4CD5>A f0_10 sf .411 .041(\) )J 296 129 :M -.235(are )A 316 129 :M -.16(d-separation )A 372 129 :M -.115(equivalent )A 421 129 :M .328 .033(if )J 436 129 :M -.313(and )A 458 129 :M .515 .052(only )J 485 129 :M -.106(if)A 59 141 :M .059(MAG\()A f4_10 sf .064(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .068(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .059(L)A f0_10 sf .15 .015(\)\) and MAG\()J f4_10 sf .064(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .068(O)A f0_10 sf (,)S f2_10 sf .039<53D5>A f0_10 sf (,)S f2_10 sf .044<4CD5>A f0_10 sf .138 .014(\)\) have the same basic colliders.)J 59 156 :M -.084(Theorem )A 98 156 :M .454 .045(1 )J 107 156 :M .694 .069(is )J 118 156 :M .218 .022(the )J 134 156 :M .337 .034(basis )J 158 156 :M .144 .014(of )J 170 156 :M .051 .005(an )J 183 156 :M -.023(O\(n)A f0_6 sf 0 -4 rm (3)S 0 4 rm f0_10 sf (m)S f0_6 sf 0 -4 rm (2)S 0 4 rm f0_10 sf (\) )S 219 156 :M .327 .033(algorithm )J 262 156 :M -.052(for )A 277 156 :M -.074(determining )A 328 156 :M -.16(d-separation )A 379 156 :M -.146(equivalence, )A 431 156 :M -.185(where )A 458 156 :M .454 .045(n )J 467 156 :M .694 .069(is )J 479 156 :M -.108(the)A 59 168 :M .489 .049(maximum number of vertices in )J f4_10 sf .216(G)A f0_6 sf 0 2 rm .09(1)A 0 -2 rm f0_10 sf .1<28>A f2_10 sf .233(O)A f0_10 sf .075(,)A f2_10 sf .166(S)A f0_10 sf .075(,)A f2_10 sf .2(L)A f0_10 sf .262 .026(\) and )J f4_10 sf .216(G)A f0_6 sf 0 2 rm .09(2)A 0 -2 rm f0_10 sf .1<28>A f2_10 sf .233(O)A f0_10 sf .075(,)A f2_10 sf .133<53D5>A f0_10 sf .075(,)A f2_10 sf .15<4CD5>A f0_10 sf .208 .021(\), )J 313 168 :M -.313(and )A 330 168 :M .656 .066(m )J 342 168 :M .694 .069(is )J 353 168 :M .218 .022(the )J 369 168 :M (number )S 403 168 :M .144 .014(of )J 415 168 :M -.074(vertices )A 449 168 :M .601 .06(in )J 461 168 :M f2_10 sf .744(O)A f0_10 sf .435 .043(. )J 476 168 :M -.273(The)A 59 180 :M .266 .027(first )J 79 180 :M .281 .028(step )J 99 180 :M .601 .06(in )J 111 180 :M -.074(determining )A 162 180 :M -.16(d-separation )A 213 180 :M -.204(equivalence )A 262 180 :M .694 .069(is )J 274 180 :M .601 .06(to )J 287 180 :M .047 .005(form )J 311 180 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 389 180 :M -.313(and )A 407 180 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .168(\)\),)A 59 192 :M .248 .025(which is O\(n)J f0_6 sf 0 -4 rm (3)S 0 4 rm f0_10 sf .224 .022(m\). Checking that the )J 206 192 :M .387 .039(two )J 225 192 :M .202 .02(MAGs )J 256 192 :M -.094(have )A 278 192 :M .218 .022(the )J 294 192 :M (same )S 318 192 :M -.182(unshielded )A 363 192 :M -.103(colliders )A 400 192 :M .694 .069(is )J 411 192 :M .071(O\(m)A f0_6 sf 0 -4 rm (3)S 0 4 rm f0_10 sf .081 .008(\), )J 442 192 :M -.313(and )A 459 192 :M -.052(for )A 474 192 :M -.438(each)A 59 204 :M .018 .002(triple of vertices all of which are adjacent to each other, there is a simple algorithm )J 395 204 :M .361 .036(that )J 414 204 :M -.137(determines )A 460 204 :M -.2(whether)A 59 216 :M -.01(there is a discriminating path that examines each edge O\(m)A f0_6 sf 0 -4 rm (2)S 0 4 rm f0_10 sf -.009(\) in the )A 330 216 :M .132 .013(MAG )J 357 216 :M .236 .024(at )J 368 216 :M .753 .075(most )J 392 216 :M .082 .008(once. )J 417 216 :M -.089(Hence, )A 448 216 :M -.037(overall )A 479 216 :M -.108(the)A 59 228 :M .278 .028(algorithm is O\(n)J f0_6 sf 0 -4 rm (3)S 0 4 rm f0_10 sf .118(m)A f0_6 sf 0 -4 rm (2)S 0 4 rm f0_10 sf .088(\).)A 252 243 :M f2_10 sf .618(Appendix)A 59 258 :M f0_10 sf -.033(For our purposes we need to represent a variety of marks attached to the ends of )A 380 258 :M -.128(edges. )A 408 258 :M .144 .014(In )J 420 258 :M -.053(general, )A 455 258 :M -.079(we )A 470 258 :M -.054(allow)A 59 270 :M -.032(that the end of an edge can be marked out of by \322)A f1_10 sf -.08A f0_10 sf -.034(\323, or can be marked with )A 366 270 :M .345 .035J 387 270 :M .144 .014(or )J 399 270 :M -.126(can )A 416 270 :M .051 .005(be )J 429 270 :M -.247(marked )A 461 270 :M .517 .052(with )J 483 270 :M -.439(an)A 59 282 :M .088 .009J 79 282 :M .144 .014(In )J 91 282 :M -.319(order )A 114 282 :M .601 .06(to )J 126 282 :M -.053(specify )A 158 282 :M .036 .004(completely )J 206 282 :M .218 .022(the )J 222 282 :M .202 .02(type )J 244 282 :M .144 .014(of )J 257 282 :M .051 .005(an )J 271 282 :M -.175(edge, )A 296 282 :M -.108(therefore, )A 338 282 :M -.079(we )A 354 282 :M -.344(need )A 376 282 :M .601 .06(to )J 389 282 :M -.053(specify )A 422 282 :M .218 .022(the )J 439 282 :M -.065(variables )A 479 282 :M -.719(and)A 59 294 :M f2_10 sf (marks)S f0_10 sf -.001( at each end. For example, the left end of "A o)A f1_10 sf S f0_10 sf ( B" can be represented as the )S 400 294 :M -.433(ordered )A 431 294 :M (pair )S 450 294 :M .349 .035([A, )J 467 294 :M .144 .014(o] )J 479 294 :M -.719(and)A 59 306 :M -.024(the right end can )A 129 306 :M .051 .005(be )J 142 306 :M -.234(represented )A 189 306 :M .144 .014(as )J 201 306 :M .218 .022(the )J 217 306 :M -.433(ordered )A 248 306 :M (pair )S 267 306 :M ([B, )S 283 306 :M .794 .079(>]. )J 299 306 :M -.188(We )A 316 306 :M .676 .068(will )J 336 306 :M .281 .028(also )J 356 306 :M .047 .005(call )J 374 306 :M .349 .035([A, )J 391 306 :M .144 .014(o] )J 403 306 :M .218 .022(the )J 419 306 :M .255 .026(A )J 430 306 :M -.313(end )A 447 306 :M .144 .014(of )J 459 306 :M .218 .022(the )J 475 306 :M -.626(edge)A 59 318 :M -.032(between A and B. The first member )A 205 318 :M .144 .014(of )J 217 318 :M .218 .022(the )J 233 318 :M -.433(ordered )A 264 318 :M (pair )S 283 318 :M .694 .069(is )J 294 318 :M -.229(called )A 320 318 :M .051 .005(an )J 333 318 :M -.062(endpoint )A 371 318 :M .144 .014(of )J 383 318 :M .051 .005(an )J 396 318 :M -.175(edge, )A 420 318 :M .758 .076(e.g. )J 439 318 :M .601 .06(in )J 451 318 :M .349 .035([A, )J 468 318 :M .144 .014(o] )J 480 318 :M -.108(the)A 59 330 :M -.009(endpoint is A. The entire edge is a set of ordered pairs representing the endpoints, e.g. {[A, o], [B, >]}. )A 473 330 :M -.145(Note)A 59 342 :M (that the edge {[B, >],[A, o]} is the same as {[A, o],[B, )S 280 342 :M -.089(>]} )A 297 342 :M (since )S 321 342 :M .786 .079(it )J 331 342 :M -.058(doesn't )A 362 342 :M (matter )S 391 342 :M .043 .004(which )J 419 342 :M -.313(end )A 436 342 :M .144 .014(of )J 448 342 :M .218 .022(the )J 464 342 :M -.344(edge )A 485 342 :M .332(is)A 59 354 :M -.022(listed first. Note that a directed edge such as A )A f1_10 sf -.057A f0_10 sf -.024( B has a mark \322)A f1_10 sf -.058A f0_10 sf -.024(\323 at the A end.)A 59 369 :M .224 .022(We say a )J f2_10 sf .114(graph)A f0_10 sf .263 .026( is an ordered triple <)J f2_10 sf .161(V)A f0_10 sf .056(,)A f2_10 sf .139(M,E)A f0_10 sf .28 .028(> where )J f2_10 sf .161(V)A f0_10 sf .262 .026( is a non-empty )J 352 369 :M .303 .03(set )J 367 369 :M .144 .014(of )J 379 369 :M -.01(vertices, )A 416 369 :M f2_10 sf .839(M)A f0_10 sf .222 .022( )J 430 369 :M .694 .069(is )J 441 369 :M .056 .006(a )J 449 369 :M -.04(non-empty)A 59 381 :M .259 .026(set of marks, and )J f2_10 sf .142(E)A f0_10 sf .168 .017( is a set of sets of )J 212 381 :M -.433(ordered )A 243 381 :M .044 .004(pairs )J 266 381 :M .144 .014(of )J 278 381 :M .218 .022(the )J 294 381 :M .047 .005(form )J 317 381 :M -.04({[V)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.045(,M)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.032(],[V)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf -.045(,M)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf -.034(]}, )A 397 381 :M -.185(where )A 424 381 :M .159(V)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 438 381 :M -.313(and )A 455 381 :M .159(V)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 469 381 :M -.235(are )A 484 381 :M .222(in)A 59 393 :M f2_10 sf .194(V)A f0_10 sf .252 .025(, V)J f0_6 sf 0 2 rm .081(1)A 0 -2 rm f0_10 sf .061 .006( )J f1_10 sf .147A f0_10 sf .217 .022( V)J f0_6 sf 0 2 rm .081(2)A 0 -2 rm f0_10 sf .306 .031(, and M)J f0_6 sf 0 2 rm .081(2)A 0 -2 rm f0_10 sf .217 .022( and )J 159 393 :M .377(M)A f0_6 sf 0 2 rm .127(2)A 0 -2 rm f0_10 sf .106 .011( )J 175 393 :M -.235(are )A 190 393 :M .601 .06(in )J 202 393 :M f2_10 sf 1.021(M)A f0_10 sf .491 .049(. )J 219 393 :M -.078(If )A 229 393 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 240 393 :M -.14(= )A 249 393 :M .655(<)A f2_10 sf .839(V)A f0_10 sf .291(,)A f2_10 sf .721(M,E)A f0_10 sf .86 .086(> )J 297 393 :M -.079(we )A 312 393 :M .133 .013(say )J 329 393 :M .361 .036(that )J 348 393 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 359 393 :M .694 .069(is )J 370 393 :M f2_10 sf .579(over)A f0_10 sf .307 .031( )J 395 393 :M f2_10 sf 1.052(V)A f0_10 sf .662 .066(. )J 410 393 :M -.25(\(Directed )A 449 393 :M -.026(graphs )A 479 393 :M -.719(and)A 59 405 :M -.023(MAGs are both special cases of graphs.\))A 59 420 :M .264 .026(In a graph, for a directed edge A )J f1_10 sf .224A f0_10 sf .204 .02( B, the edge is )J f2_10 sf .351 .035(out of)J f0_10 sf .179 .018( A and )J f2_10 sf .095(into)A f0_10 sf .057 .006( )J 341 420 :M .275 .028(B, )J 354 420 :M -.313(and )A 371 420 :M .255 .026(A )J 382 420 :M .694 .069(is )J 393 420 :M f2_10 sf .333(parent)A f0_10 sf .177 .018( )J 427 420 :M .144 .014(of )J 439 420 :M -.17(B )A 449 420 :M -.313(and )A 466 420 :M -.17(B )A 476 420 :M .694 .069(is )J 487 420 :M (a)S 59 432 :M f2_10 sf .606(child)A f0_10 sf .358 .036( )J 87 432 :M .144 .014(of )J 99 432 :M .65 .065(A. )J 113 432 :M .255 .026(A )J 124 432 :M -.268(sequence )A 162 432 :M .144 .014(of )J 174 432 :M -.253(edges )A 199 432 :M .463( )J 248 432 :M .601 .06(in )J 260 432 :M f4_10 sf .209(G)A f0_10 sf .072 .007( )J 271 432 :M .694 .069(is )J 282 432 :M .051 .005(an )J 295 432 :M f2_10 sf 2.202 .22(undirected )J 349 432 :M 2.182 .218(path )J 375 432 :M f0_10 sf .328 .033(if )J 385 432 :M -.313(and )A 402 432 :M .515 .052(only )J 424 432 :M .328 .033(if )J 435 432 :M -.097(there )A 459 432 :M .454 .045(exists )J 487 432 :M (a)S 59 444 :M .329 .033(sequence of vertices such that for 1 )J cF f1_10 sf .016A sf .159 .016( i )J cF f1_10 sf .016A sf .159 .016( n E)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .263 .026( has endpoints V)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .186 .019( and V)J f0_6 sf 0 2 rm .049(i+1)A 0 -2 rm f0_10 sf .188 .019(, and E)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( )S f1_10 sf .099A f0_10 sf .13 .013( E)J f0_6 sf 0 2 rm .049(i+1)A 0 -2 rm f0_10 sf .082 .008(. )J 463 444 :M .255 .026(A )J 474 444 :M -.072(path)A 59 456 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 70 456 :M .694 .069(is )J 81 456 :M f2_10 sf .74(acyclic)A f0_10 sf .448 .045( )J 119 456 :M .328 .033(if )J 129 456 :M .417 .042(no )J 144 456 :M -.081(vertex )A 173 456 :M -.147(appears )A 207 456 :M (more )S 232 456 :M .202 .02(than )J 254 456 :M -.094(once )A 277 456 :M .601 .06(in )J 290 456 :M .218 .022(the )J 307 456 :M -.131(corresponding )A 367 456 :M -.268(sequence )A 406 456 :M .144 .014(of )J 419 456 :M -.01(vertices. )A 457 456 :M -.188(We )A 475 456 :M .149(will)A 59 468 :M .041 .004(assume )J 92 468 :M .361 .036(that )J 111 468 :M .051 .005(an )J 124 468 :M -.27(undirected )A 168 468 :M .202 .02(path )J 190 468 :M .694 .069(is )J 202 468 :M -.116(acyclic )A 234 468 :M .315 .032(unless )J 264 468 :M -.049(specifically )A 314 468 :M -.079(mentioned )A 360 468 :M .065 .007(otherwise. )J 406 468 :M .255 .026(A )J 418 468 :M -.268(sequence )A 457 468 :M .144 .014(of )J 470 468 :M -.442(edges)A 59 480 :M .394( in )J f4_10 sf .484(G)A f0_10 sf .356 .036( is a )J 146 480 :M f2_10 sf 1.957 .196(directed )J 188 480 :M 2.182 .218(path )J 214 480 :M f4_10 sf .951(D)A f2_10 sf .329 .033( )J 226 480 :M .538(from)A f0_10 sf .255 .025( )J 253 480 :M .159(V)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 267 480 :M .601 .06(to )J 279 480 :M .727(V)A f0_6 sf 0 2 rm .302(n)A 0 -2 rm f2_10 sf .252 .025( )J 294 480 :M f0_10 sf .328 .033(if )J 304 480 :M -.313(and )A 321 480 :M .515 .052(only )J 343 480 :M .328 .033(if )J 353 480 :M -.097(there )A 376 480 :M .454 .045(exists )J 403 480 :M .056 .006(a )J 411 480 :M -.268(sequence )A 449 480 :M .144 .014(of )J 461 480 :M -.156(vertices)A 59 492 :M .144( such that for 1 )J cF f1_10 sf .023A sf .234 .023( i )J cF f1_10 sf .023A sf .234 .023( n, there is a directed edge V)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .051 .005( )J f1_10 sf .221A f0_10 sf .181 .018( V)J f0_6 sf 0 2 rm .06(i+1)A 0 -2 rm f0_10 sf .166 .017( on D. If )J 384 492 :M -.097(there )A 407 492 :M .694 .069(is )J 418 492 :M .051 .005(an )J 431 492 :M -.116(acyclic )A 462 492 :M -.457(directed)A 59 504 :M .154 .015(path from A )J 112 504 :M .601 .06(to )J 124 504 :M -.17(B )A 134 504 :M .144 .014(or )J 146 504 :M -.17(B )A 156 504 :M -.14(= )A 165 504 :M .255 .026(A )J 176 504 :M .202 .02(then )J 197 504 :M .255 .026(A )J 208 504 :M .694 .069(is )J 219 504 :M .051 .005(an )J 232 504 :M f2_10 sf .515(ancestor)A f0_10 sf .285 .029( )J 276 504 :M .144 .014(of )J 288 504 :M .275 .028(B, )J 301 504 :M -.313(and )A 318 504 :M -.17(B )A 328 504 :M .694 .069(is )J 339 504 :M .056 .006(a )J 347 504 :M f2_10 sf .355(descendant)A f0_10 sf .186 .019( )J 402 504 :M .144 .014(of )J 414 504 :M .65 .065(A. )J 428 504 :M -.078(If )A 438 504 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 449 504 :M .694 .069(is )J 460 504 :M .056 .006(a )J 468 504 :M .303 .03(set )J 483 504 :M -.328(of)A 59 516 :M -.009(variables, )A 101 516 :M .255 .026(A )J 112 516 :M .694 .069(is )J 123 516 :M .051 .005(an )J 136 516 :M f2_10 sf .515(ancestor)A f0_10 sf .285 .029( )J 180 516 :M .144 .014(of )J 192 516 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 203 516 :M .328 .033(if )J 213 516 :M -.313(and )A 230 516 :M .515 .052(only )J 252 516 :M .328 .033(if )J 262 516 :M .786 .079(it )J 272 516 :M .694 .069(is )J 283 516 :M .051 .005(an )J 297 516 :M -.101(ancestor )A 334 516 :M .144 .014(of )J 347 516 :M .056 .006(a )J 356 516 :M -.044(member )A 393 516 :M .144 .014(of )J 406 516 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(, )J 421 516 :M -.313(and )A 439 516 :M .502 .05(similarly )J 480 516 :M -.328(for)A 59 528 :M f2_10 sf .253(descendant)A f0_10 sf .3 .03(. If )J f2_10 sf .383(X)A f0_10 sf .485 .049( is a set of vertices in )J f4_10 sf .383(G)A f0_10 sf .357 .036(, let )J f2_10 sf .249(Ancestors)A f0_10 sf .177<28>A f2_10 sf .383(X)A f0_10 sf .466 .047(\) be the set of all )J 379 528 :M -.078(ancestors )A 419 528 :M .144 .014(of )J 431 528 :M -.021(members )A 471 528 :M .144 .014(of )J 483 528 :M f2_10 sf (X)S 59 540 :M f0_10 sf .232 .023(in )J f4_10 sf .212(G)A f0_10 sf .098<28>A f2_10 sf .229(O)A f0_10 sf .073(,)A f2_10 sf .163(S)A f0_10 sf .073(,)A f2_10 sf .196(L)A f0_10 sf .265 .027(\). A vertex V is a )J f2_10 sf .118(collider)A f0_10 sf .368 .037( on an undirected path )J f4_10 sf .212(U)A f0_10 sf .273 .027( if and only )J 365 540 :M .328 .033(if )J 375 540 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 386 540 :M .098 .01(contains )J 423 540 :M .056 .006(a )J 431 540 :M (pair )S 450 540 :M .144 .014(of )J 462 540 :M -.063(distinct)A 59 552 :M -.03(edges adjacent on the path and into V. The )A 232 552 :M f2_10 sf .63(orientation)A f0_10 sf .367 .037( )J 290 552 :M .144 .014(of )J 302 552 :M .051 .005(an )J 315 552 :M -.116(acyclic )A 346 552 :M -.27(undirected )A 389 552 :M .202 .02(path )J 410 552 :M -.116(between )A 446 552 :M .255 .026(A )J 457 552 :M -.313(and )A 474 552 :M -.17(B )A 484 552 :M .332(is)A 59 564 :M .218 .022(the )J 75 564 :M .303 .03(set )J 90 564 :M .474 .047(consisting )J 135 564 :M .144 .014(of )J 147 564 :M .218 .022(the )J 163 564 :M .255 .026(A )J 174 564 :M -.313(end )A 191 564 :M .144 .014(of )J 203 564 :M .218 .022(the )J 219 564 :M -.344(edge )A 240 564 :M .417 .042(on )J 254 564 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 265 564 :M .361 .036(that )J 284 564 :M .098 .01(contains )J 321 564 :M .65 .065(A, )J 335 564 :M -.313(and )A 352 564 :M .218 .022(the )J 368 564 :M -.17(B )A 379 564 :M -.313(end )A 397 564 :M .144 .014(of )J 410 564 :M .218 .022(the )J 427 564 :M -.344(edge )A 449 564 :M .2(on)A f4_10 sf .1 .01( )J 464 564 :M .209(U)A f0_10 sf .072 .007( )J 476 564 :M (that)S 59 576 :M -.011(contains B. Say that a vertex V on an undirected path )A f4_10 sf (U)S f0_10 sf -.013( between X )A 330 576 :M -.313(and )A 347 576 :M .255 .026(Y )J 358 576 :M .694 .069(is )J 369 576 :M f2_10 sf .684(active)A f0_10 sf .411 .041( )J 402 576 :M .417 .042(on )J 416 576 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 427 576 :M .189 .019(given )J 453 576 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 464 576 :M .134<28>A f2_10 sf .268(Z)A f0_10 sf .1 .01( )J 478 576 :M .111(not)A 59 588 :M .125 .012(containing X and Y\) if and only if either V is not a collider on )J f4_10 sf .077(U)A f0_10 sf .092 .009( and not in )J f2_10 sf .071(Z)A f0_10 sf .049 .005(, )J 378 588 :M .144 .014(or )J 390 588 :M .255 .026(V )J 401 588 :M .694 .069(is )J 412 588 :M .056 .006(a )J 420 588 :M -.13(collider )A 453 588 :M .417 .042(on )J 467 588 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 478 588 :M -.719(and)A 59 600 :M .262 .026(is an ancestor of )J f2_10 sf .149(Z)A f0_10 sf .294 .029(. For three disjoint sets of variables )J f2_10 sf .162(A)A f0_10 sf .093 .009(, )J f2_10 sf .149(B)A f0_10 sf .102 .01(, )J 307 600 :M -.313(and )A 324 600 :M f2_10 sf 1.052(C)A f0_10 sf .662 .066(, )J 339 600 :M f2_10 sf .951(A)A f0_10 sf .329 .033( )J 351 600 :M .694 .069(is )J 362 600 :M f2_10 sf .421(d-connected)A f0_10 sf .224 .022( )J 422 600 :M .601 .06(to )J 434 600 :M f2_10 sf .604(B)A f0_10 sf .226 .023( )J 445 600 :M .189 .019(given )J 471 600 :M f2_10 sf .951(C)A f0_10 sf .329 .033( )J 483 600 :M .222(in)A 59 612 :M .311 .031(graph )J f4_10 sf .142(G)A f0_10 sf .112 .011(, if )J 107 612 :M -.313(and )A 124 612 :M .515 .052(only )J 146 612 :M .328 .033(if )J 156 612 :M -.097(there )A 179 612 :M .694 .069(is )J 190 612 :M .051 .005(an )J 203 612 :M -.27(undirected )A 246 612 :M .202 .02(path )J 267 612 :M .047 .005(from )J 290 612 :M .281 .028(some )J 315 612 :M -.044(member )A 351 612 :M .144 .014(of )J 363 612 :M f2_10 sf .951(A)A f0_10 sf .329 .033( )J 375 612 :M .601 .06(to )J 387 612 :M .056 .006(a )J 395 612 :M -.044(member )A 431 612 :M .144 .014(of )J 443 612 :M f2_10 sf .604(B)A f0_10 sf .226 .023( )J 454 612 :M .123 .012(such )J 476 612 :M (that)S 59 624 :M .072 .007(every vertex on )J f4_10 sf (U)S f0_10 sf .026 .003( is )J 144 624 :M -.062(active )A 171 624 :M .189 .019(given )J 197 624 :M f2_10 sf .868(C)A f0_10 sf .577 .058(; )J 212 624 :M -.052(for )A 227 624 :M -.097(three )A 250 624 :M .276 .028(disjoint )J 284 624 :M .361 .036(sets )J 303 624 :M .144 .014(of )J 315 624 :M -.065(variables )A 354 624 :M f2_10 sf 1.052(A)A f0_10 sf .662 .066(, )J 369 624 :M f2_10 sf .76(B)A f0_10 sf .518 .052(, )J 383 624 :M -.313(and )A 400 624 :M f2_10 sf 1.052(C)A f0_10 sf .662 .066(, )J 415 624 :M f2_10 sf .951(A)A f0_10 sf .329 .033( )J 427 624 :M .694 .069(is )J 438 624 :M f2_10 sf .246(d-separated)A 59 636 :M f0_10 sf .316 .032(from )J f2_10 sf .144(B)A f0_10 sf .218 .022( given )J f2_10 sf .156(C)A f0_10 sf .206 .021( in graph )J f4_10 sf .156(G)A f0_10 sf .204 .02(, if and only )J f2_10 sf .156(A)A f0_10 sf .263 .026( is not d-connected to )J f2_10 sf .144(B)A f0_10 sf .218 .022( given )J f2_10 sf .156(C)A f0_10 sf (.)S 59 651 :M .144 .014(In )J 71 651 :M .056 .006(a )J 79 651 :M f2_10 sf 1.957 .196(directed )J 121 651 :M .409(graph)A f0_10 sf .363 .036(, )J 155 651 :M .388 .039(all )J 169 651 :M .144 .014(of )J 181 651 :M .218 .022(the )J 197 651 :M -.253(edges )A 222 651 :M -.235(are )A 237 651 :M -.337(directed )A 270 651 :M -.128(edges. )A 298 651 :M .255 .026(A )J 310 651 :M -.337(directed )A 344 651 :M -.053(graph )A 371 651 :M .694 .069(is )J 383 651 :M f2_10 sf .74(acyclic)A f0_10 sf .448 .045( )J 422 651 :M .328 .033(if )J 433 651 :M -.313(and )A 451 651 :M .515 .052(only )J 474 651 :M .328 .033(if )J 485 651 :M .443(it)A 59 663 :M .098 .01(contains )J 96 663 :M .417 .042(no )J 110 663 :M -.337(directed )A 143 663 :M -.062(cyclic )A 170 663 :M .559 .056(paths. )J 198 663 :M (Lemma )S 232 663 :M .454 .045(1 )J 241 663 :M .694 .069(is )J 252 663 :M .056 .006(a )J 260 663 :M .523 .052(simple )J 291 663 :M -.081(generalization )A 350 663 :M .144 .014(of )J 362 663 :M (Lemma )S 397 663 :M 1 .1(3.3.1 )J 423 663 :M .601 .06(in )J 436 663 :M .429 .043(Spirtes )J 469 663 :M f4_10 sf .236 .024(et )J 481 663 :M -.139(al.)A 59 675 :M f0_10 sf -.026(\(1993\).)A endp %%Page: 4 4 %%BeginPageSetup initializepage (peter; page: 4 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (4)S gR gS 0 0 552 730 rC 59 51 :M f2_10 sf .296 .03(Lemma 1:)J f0_10 sf .152 .015( In a directed acyclic graph )J f4_10 sf .088(G)A f0_10 sf .127 .013( over a set of vertices )J f2_10 sf .088(V)A f0_10 sf .195 .02(, if the following conditions hold:)J 77 66 :M .37 .037(\(a\) )J f4_10 sf .233(R)A f0_10 sf .428 .043( is a sequence of vertices in )J f2_10 sf .275(V)A f0_10 sf .334 .033( from A to B, )J f4_10 sf .233(R)A f0_10 sf .087 .009( )J f1_10 sf .253 .025J f0_10 sf .245(A f0_10 sf .275(X)A f0_6 sf 0 2 rm .114(0)A 0 -2 rm f0_10 sf .25<2CC958>A f0_6 sf 0 2 rm .119(n+1)A 0 -2 rm f1_10 sf .209A f0_10 sf .481 .048(B>, such that )J f1_10 sf .271(")A f0_10 sf .554 .055<692CCA30CA>J cF f1_10 sf .055A sf .554 .055( )J 453 66 :M .656 .066(i )J 460 66 :M cF f1_10 sf .092A sf .92 .092( )J 470 66 :M .833 .083(n, )J 482 66 :M .092(X)A f0_6 sf 0 2 rm (i)S 0 -2 rm 77 78 :M f1_10 sf .055A f0_10 sf .081 .008( X)J f0_6 sf 0 2 rm .027(i+1)A 0 -2 rm f0_10 sf .103 .01( \(the X)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .094 .009( are only )J f4_10 sf .246 .025(pairwise distinct)J f0_10 sf .144 .014( , i.e. not necessarily distinct\),)J 77 93 :M .176<286229CA>A f2_10 sf .381 .038(Z )J f1_10 sf .355A f0_10 sf .113 .011( )J f2_10 sf .36(V)A f0_10 sf .26(\\{A,B},)A 77 108 :M (\(c\) )S f2_10 sf (T)S f0_10 sf -.01( is a set of undirected paths such that)A 95 123 :M .224 .022(\(i\)\312for each pair )J 162 123 :M .144 .014(of )J 174 123 :M -.064(consecutive )A 224 123 :M -.074(vertices )A 258 123 :M .601 .06(in )J 270 123 :M f4_10 sf 1.04(R)A f0_10 sf .774 .077(, )J 284 123 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 297 123 :M -.313(and )A 314 123 :M .26(X)A f0_6 sf 0 2 rm .097(i+1)A 0 -2 rm f0_10 sf .164 .016(, )J 336 123 :M -.097(there )A 359 123 :M .694 .069(is )J 370 123 :M .056 .006(a )J 378 123 :M -.119(unique )A 408 123 :M -.27(undirected )A 451 123 :M .202 .02(path )J 472 123 :M .601 .06(in )J 484 123 :M f2_10 sf (T)S 95 135 :M f0_10 sf -.007(that d-connects X)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( and X)S f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf ( given )S f2_10 sf (Z)S f0_10 sf (\\{X)S f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( , X)S f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf (},)S 95 150 :M .832 .083(\(ii\)\312if some vertex X)J f0_6 sf 0 2 rm .156(k)A 0 -2 rm f0_10 sf .277 .028( in )J f4_10 sf .318(R)A f0_10 sf .308 .031( is in )J f2_10 sf .347(Z)A f0_10 sf .494 .049(, then )J 261 150 :M .218 .022(the )J 277 150 :M .263 .026(paths )J 302 150 :M .601 .06(in )J 314 150 :M f2_10 sf .604(T)A f0_10 sf .226 .023( )J 325 150 :M .361 .036(that )J 344 150 :M .039 .004(contain )J 377 150 :M .159(X)A f0_6 sf 0 2 rm .066(k)A 0 -2 rm f0_10 sf .055 .006( )J 391 150 :M .144 .014(as )J 403 150 :M .051 .005(an )J 416 150 :M -.062(endpoint )A 454 150 :M -.102(collide )A 484 150 :M -.217(at)A 95 162 :M (X)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.004(, \(i.e. all such paths are directed into X)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf <29>S 95 177 :M .25 .025(\(iii\)\312if for three vertices )J 194 177 :M .266(X)A f0_6 sf 0 2 rm .11<6BD031>A 0 -2 rm f0_10 sf .167 .017(, )J 217 177 :M .37(X)A f0_6 sf 0 2 rm .154(k)A 0 -2 rm f0_10 sf .233 .023(, )J 234 177 :M (X)S f0_6 sf 0 2 rm -.017(k+1)A 0 -2 rm f0_10 sf ( )S 254 177 :M -.09(occurring )A 295 177 :M .601 .06(in )J 307 177 :M f4_10 sf 1.04(R)A f0_10 sf .774 .077(, )J 321 177 :M .218 .022(the )J 337 177 :M -.142(d-connecting )A 391 177 :M .263 .026(paths )J 416 177 :M .601 .06(in )J 428 177 :M f2_10 sf .604(T)A f0_10 sf .226 .023( )J 439 177 :M -.116(between )A 475 177 :M -.098(X)A f0_6 sf 0 2 rm -.061<6BD031>A 0 -2 rm 95 189 :M f0_10 sf -.037(and X)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.032(, and X)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.034( and X)A f0_6 sf 0 2 rm -.024(k+1)A 0 -2 rm f0_10 sf -.028(, collide at X)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.032( then X)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.03( has a descendant in )A f2_10 sf -.051(Z)A f0_10 sf (,)S 59 204 :M .072 .007(then there is a path )J f4_10 sf (U)S f0_10 sf .035 .003( in )J f4_10 sf .053 .005(G )J f0_10 sf .125 .013(that d-connects A)J f1_10 sf S f0_10 sf (X)S f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .066 .007( and B)J f1_10 sf S f0_10 sf (X)S f0_6 sf 0 2 rm .021(n+1)A 0 -2 rm f0_10 sf .066 .007( given )J f2_10 sf (Z)S f0_10 sf .052 .005( that )J 358 204 :M .098 .01(contains )J 395 204 :M .515 .052(only )J 417 204 :M -.253(edges )A 442 204 :M -.09(occurring )A 483 204 :M .222(in)A 59 216 :M f2_10 sf .604(T)A f0_10 sf (.)S 59 231 :M f4_10 sf .469(U)A f0_10 sf .414 .041( is an )J f2_10 sf 1.671 .167(inducing path)J f0_10 sf .78 .078( between X )J 207 231 :M -.313(and )A 224 231 :M .255 .026(Y )J 235 231 :M .601 .06(in )J 247 231 :M f4_10 sf .513(G)A f0_10 sf .236<28>A f2_10 sf .553(O)A f0_10 sf .178(,)A f2_10 sf .395(S)A f0_10 sf .178(,)A f2_10 sf .474(L)A f0_10 sf .493 .049(\), )J 295 231 :M .328 .033(if )J 305 231 :M -.313(and )A 322 231 :M .515 .052(only )J 344 231 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 355 231 :M .694 .069(is )J 366 231 :M .051 .005(an )J 379 231 :M -.116(acyclic )A 410 231 :M -.27(undirected )A 453 231 :M .202 .02(path )J 474 231 :M -.109(such)A 59 243 :M .133 .013(that every member of )J f2_10 sf .071(O)A f0_10 sf ( )S f1_10 sf .071A f0_10 sf ( )S f2_10 sf .051(S)A f0_10 sf .057 .006( on )J f4_10 sf .066(U)A f0_10 sf .087 .009( is a collider on )J f4_10 sf .066(U)A f0_10 sf .114 .011(, and every collider on )J f4_10 sf .066(U)A f0_10 sf .095 .009( is an ancestor of )J 441 243 :M -.208({X,Y} )A 470 243 :M f1_10 sf .62A f0_10 sf .202 .02( )J 482 243 :M f2_10 sf 1.339(S)A f0_10 sf (.)S 59 255 :M -.014(\(This is a generalization of the concept of inducing path that was introduced in )A 376 255 :M -.14(Verma )A 406 255 :M -.313(and )A 423 255 :M (Pearl )S 447 255 :M .42 .042(1990\). )J 477 255 :M -.273(The)A 59 267 :M .014 .001(following sequence of lemmas state that for every subset )J f2_10 sf (W)S f0_10 sf ( of )S f2_10 sf (O)S f0_10 sf .011 .001(, X and Y are d-connected given )J f2_10 sf (W)S f0_10 sf ( )S f1_10 sf S f0_10 sf ( )S f2_10 sf (S)S f0_10 sf ( )S 484 267 :M .222(in)A 59 279 :M f4_10 sf .077(G)A f0_10 sf <28>S f2_10 sf .083(O)A f0_10 sf (,)S f2_10 sf .059(S)A f0_10 sf (,)S f2_10 sf .071(L)A f0_10 sf .126 .013(\) if and only there is an inducing path between X and Y in )J f4_10 sf .077(G)A f0_10 sf <28>S f2_10 sf .083(O)A f0_10 sf (,)S f2_10 sf .059(S)A f0_10 sf (,)S f2_10 sf .071(L)A f0_10 sf .134 .013(\). For space reasons we do )J 479 279 :M .111(not)A 59 291 :M .059 .006(present the proofs here, but they are simple modifications of the proofs that )J 365 291 :M -.19(appear )A 394 291 :M .601 .06(in )J 406 291 :M .429 .043(Spirtes )J 438 291 :M .236 .024(et )J 449 291 :M .602 .06(al. )J 463 291 :M -.026(\(1993\),)A 59 303 :M .032 .003(in which the case of latent variables without selection bias is considered. \(There is no analog of Lemma 4 in)J 59 315 :M .275 .028(Spirtes )J f4_10 sf .204 .02(et al.)J f0_10 sf .211 .021( 1993, but the proof is very similar to that of Lemma 2 and Lemma 3.\))J 59 330 :M f2_10 sf .431 .043(Lemma 2)J f0_10 sf .231 .023(: In directed graph )J f4_10 sf .132(G)A f0_10 sf .061<28>A f2_10 sf .142(O)A f0_10 sf (,)S f2_10 sf .101(S)A f0_10 sf (,)S f2_10 sf .122(L)A f0_10 sf .235 .023(\), if there is an inducing path between )J 371 330 :M .255 .026(A )J 382 330 :M -.313(and )A 399 330 :M -.17(B )A 409 330 :M .361 .036(that )J 428 330 :M .694 .069(is )J 439 330 :M .555 .056(out )J 456 330 :M .144 .014(of )J 468 330 :M .255 .026(A )J 479 330 :M -.719(and)A 59 342 :M .032 .003(into B, then for any subset )J f2_10 sf (Z)S f0_10 sf ( of )S f2_10 sf (O)S f0_10 sf .039 .004(\\{A,B} there is an undirected path )J f4_10 sf (C)S f0_10 sf ( )S 345 342 :M .361 .036(that )J 364 342 :M -.181(d-connects )A 409 342 :M .255 .026(A )J 420 342 :M -.313(and )A 437 342 :M -.17(B )A 447 342 :M .189 .019(given )J 473 342 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 484 342 :M f1_10 sf S 59 354 :M f2_10 sf .208(S)A f0_10 sf .378 .038( that is out of A and into B.)J 59 369 :M f2_10 sf .514 .051(Lemma 3:)J f0_10 sf .103 .01( If )J f4_10 sf .153(G)A f0_10 sf .071<28>A f2_10 sf .165(O)A f0_10 sf .053(,)A f2_10 sf .118(S)A f0_10 sf .053(,)A f2_10 sf .141(L)A f0_10 sf .261 .026(\) is a directed acyclic graph, and there is )J 320 369 :M .051 .005(an )J 333 369 :M -.062(inducing )A 371 369 :M .202 .02(path )J 392 369 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 403 369 :M -.116(between )A 439 369 :M .255 .026(A )J 450 369 :M -.313(and )A 467 369 :M -.17(B )A 477 369 :M (that)S 59 381 :M .02 .002(is into A and into B then for every subset )J f2_10 sf (Z)S f0_10 sf ( of )S f2_10 sf (O)S f0_10 sf .026 .003(\\{A,B} there is an undirected path )J f4_10 sf (C)S f0_10 sf ( )S 404 381 :M .361 .036(that )J 423 381 :M -.181(d-connects )A 468 381 :M .255 .026(A )J 479 381 :M -.719(and)A 59 393 :M .462 .046(B given )J f2_10 sf .255(Z)A f0_10 sf .087 .009( )J f1_10 sf .293A f0_10 sf .087 .009( )J f2_10 sf .212(S)A f0_10 sf .402 .04( that is into A and into B.)J 59 408 :M f2_10 sf .514 .051(Lemma 4:)J f0_10 sf .103 .01( If )J f4_10 sf .153(G)A f0_10 sf .071<28>A f2_10 sf .165(O)A f0_10 sf .053(,)A f2_10 sf .118(S)A f0_10 sf .053(,)A f2_10 sf .141(L)A f0_10 sf .261 .026(\) is a directed acyclic graph, and there is )J 320 408 :M .051 .005(an )J 333 408 :M -.062(inducing )A 371 408 :M .202 .02(path )J 392 408 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 403 408 :M -.116(between )A 439 408 :M .255 .026(A )J 450 408 :M -.313(and )A 467 408 :M -.17(B )A 477 408 :M (that)S 59 420 :M -.004(is out of A and out of B then for every subset )A f2_10 sf (Z)S f0_10 sf ( of )S f2_10 sf (O)S f0_10 sf -.004(\\{A,B} there is an undirected path )A 410 420 :M f4_10 sf .604(C)A f0_10 sf .226 .023( )J 421 420 :M .361 .036(that )J 440 420 :M -.181(d-connects )A 485 420 :M (A)S 59 432 :M .428 .043(and B given )J f2_10 sf .236(Z)A f0_10 sf .08 .008( )J f1_10 sf .272A f0_10 sf .08 .008( )J f2_10 sf .197(S)A f0_10 sf (.)S 59 447 :M f2_10 sf .143 .014(Lemma 5: )J f0_10 sf .063 .006(If )J f4_10 sf .065(G)A f0_10 sf <28>S f2_10 sf .07(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .06(L)A f0_10 sf .121 .012(\) is a directed acyclic graph and an undirected path )J f4_10 sf .065(U)A f0_10 sf .048 .005( in )J f4_10 sf .065(G)A f0_10 sf <28>S f2_10 sf .07(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .06(L)A f0_10 sf .145 .014(\) d-connects A and)J 59 459 :M .325 .032(B given \(\()J f2_10 sf .113(Ancestors)A f0_10 sf .45 .045(\({A,B} )J f1_10 sf .185A f0_10 sf .06A f2_10 sf .134(S)A f0_10 sf .08<29>A f1_10 sf .204 .02<20C7>J f0_10 sf .055 .005( )J f2_10 sf .187(O)A f0_10 sf .117 .012(\) )J f1_10 sf .185A f0_10 sf .055 .005( )J f2_10 sf .134(S)A f0_10 sf .396 .04(\)\\{A,B} then )J f4_10 sf .173(U)A f0_10 sf .316 .032( is an inducing path between A and B.)J 59 474 :M .071 .007(The following lemma follows from a simple application of d-separation to discriminating paths.)J 59 489 :M f2_10 sf .62 .062(Lemma 6: )J f0_10 sf .861 .086(In MAG\()J f4_10 sf .282(G)A f0_10 sf .13<28>A f2_10 sf .304(O)A f0_10 sf .098(,)A f2_10 sf .217(S)A f0_10 sf .098(,)A f2_10 sf .26(L)A f0_10 sf .293 .029(\)\), if )J f4_10 sf .282(U)A f0_10 sf .471 .047( is a discriminating path for B )J 339 489 :M -.116(between )A 375 489 :M .255 .026(X )J 386 489 :M -.313(and )A 403 489 :M .65 .065(Y, )J 417 489 :M -.313(and )A 434 489 :M -.17(B )A 444 489 :M .694 .069(is )J 455 489 :M .056 .006(a )J 463 489 :M -.22(collider)A 59 501 :M .168 .017(on )J f4_10 sf .126(U)A f0_10 sf .148 .015( then B )J 112 501 :M .694 .069(is )J 123 501 :M .417 .042(no )J 137 501 :M .303 .03(set )J 152 501 :M .361 .036(that )J 171 501 :M -.224(d-separates )A 217 501 :M .255 .026(X )J 228 501 :M -.313(and )A 245 501 :M .65 .065(Y, )J 259 501 :M -.313(and )A 276 501 :M .328 .033(if )J 286 501 :M -.17(B )A 296 501 :M .694 .069(is )J 307 501 :M .555 .056(not )J 324 501 :M .056 .006(a )J 332 501 :M -.13(collider )A 365 501 :M .417 .042(on )J 379 501 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 393 501 :M -.17(B )A 403 501 :M .694 .069(is )J 414 501 :M .601 .06(in )J 426 501 :M -.141(every )A 451 501 :M .303 .03(set )J 466 501 :M .361 .036(that )J 485 501 :M -1.328(d-)A 59 513 :M -.089(separates X and Y.)A 59 528 :M .046 .005(Note )J 82 528 :M .361 .036(that )J 101 528 :M .786 .079(it )J 111 528 :M .298 .03(follows )J 145 528 :M -.13(directly )A 178 528 :M .047 .005(from )J 201 528 :M .218 .022(the )J 217 528 :M -.038(definition )A 259 528 :M .144 .014(of )J 271 528 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 345 528 :M .361 .036(that )J 364 528 :M -.097(there )A 388 528 :M -.235(are )A 404 528 :M .417 .042(no )J 419 528 :M -.253(edges )A 445 528 :M .255 .026(A )J 457 528 :M (o)S f1_10 sf -.07A f0_10 sf -.064(B )A 483 528 :M .222(in)A 59 540 :M .058(MAG\()A f4_10 sf .063(G)A f0_10 sf <28>S f2_10 sf .068(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .058(L)A f0_10 sf .092 .009(\)\), and if there is an edge A o)J f1_10 sf .087A f0_10 sf .118 .012(* B in MAG\(\()J f4_10 sf .063(G)A f0_10 sf <28>S f2_10 sf .068(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .058(L)A f0_10 sf .11 .011(\)\), then the A endpoint of every )J 474 540 :M -.626(edge)A 59 552 :M .418 .042(in MAG\()J f4_10 sf .139(G)A f0_10 sf .064<28>A f2_10 sf .15(O)A f0_10 sf (,)S f2_10 sf .107(S)A f0_10 sf (,)S f2_10 sf .128(L)A f0_10 sf .227 .023(\)\) is \322o\323. Hence if A is a collider on any path in MAG\()J f4_10 sf .139(G)A f0_10 sf .064<28>A f2_10 sf .15(O)A f0_10 sf (,)S f2_10 sf .107(S)A f0_10 sf (,)S f2_10 sf .128(L)A f0_10 sf .188 .019(\)\), the )J 420 552 :M .255 .026(A )J 431 552 :M -.062(endpoint )A 469 552 :M .144 .014(of )J 481 552 :M (no)S 59 564 :M .201 .02(edge in MAG\()J f4_10 sf .08(G)A f0_10 sf <28>S f2_10 sf .086(O)A f0_10 sf (,)S f2_10 sf .061(S)A f0_10 sf (,)S f2_10 sf .074(L)A f0_10 sf .112 .011(\)\) is a \322o\323.)J 59 579 :M .121 .012(Let l\()J f4_10 sf .064(U)A f0_10 sf .041(,C)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf (,)S f2_10 sf .059(Z)A f0_10 sf .119 .012(\) be the length of a shortest directed path from collider C)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .055 .006( on )J f4_10 sf .064(U)A f0_10 sf .049 .005( to )J 376 579 :M .056 .006(a )J 384 579 :M -.044(member )A 420 579 :M .144 .014(of )J 432 579 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(. )J 446 579 :M .134 .013(Let )J 463 579 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 474 579 :M .051 .005(be )J 487 579 :M (a)S 59 591 :M f2_10 sf 1.646 .165(minimal d-connecting path)J f0_10 sf .771 .077( between X and Y given )J f2_10 sf .43(Z)A f0_10 sf .556 .056( if and only if )J f4_10 sf .465(U)A f0_10 sf .327 .033( is )J 376 591 :M .056 .006(a )J 384 591 :M -.142(d-connecting )A 438 591 :M .202 .02(path )J 459 591 :M -.219(between)A 59 603 :M .018 .002(X and Y given )J f2_10 sf (Z)S f0_10 sf .017 .002(, and there is no other path )J f4_10 sf (V)S f0_10 sf .022 .002( d-connecting X and )J 327 603 :M .255 .026(Y )J 338 603 :M .189 .019(given )J 364 603 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 375 603 :M .123 .012(such )J 397 603 :M .361 .036(that )J 416 603 :M -.044(either )A 442 603 :M f4_10 sf .987(V)A f0_10 sf .404 .04( )J 453 603 :M .133 .013(has )J 470 603 :M -.438(fewer)A 59 615 :M .219 .022(edges than )J f4_10 sf .109(U)A f0_10 sf .096 .01(, or )J f4_10 sf .092(V)A f0_10 sf .182 .018( has the same number of edges as )J f4_10 sf .109(U)A f0_10 sf .152 .015( and the sum over j of l\()J f4_10 sf .092(V)A f0_10 sf .073(,D)A f0_6 sf 0 2 rm (j)S 0 -2 rm f0_10 sf (,)S f2_10 sf .101(Z)A f0_10 sf .146 .015(\) is less than )J 458 615 :M .218 .022(the )J 474 615 :M .166(sum)A 59 627 :M .202 .02(over i of l\()J f4_10 sf .144(U)A f0_10 sf .092(,C)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf (,)S f2_10 sf .133(Z)A f0_10 sf .277 .028(\). Say that two undirected paths )J f4_10 sf .144(U)A f0_10 sf .155 .016( and )J f4_10 sf .105<55D5>A f0_10 sf .244 .024( which contain a vertex C )J f2_10 sf .089(disagree)A f0_10 sf .13 .013( at C if C )J 484 627 :M .332(is)A 59 639 :M .054 .005(a collider on )J f4_10 sf (U)S f0_10 sf .04 .004( but not on )J f4_10 sf <55D5>S f0_10 sf .077 .008(, or vice-versa.)J 59 654 :M f2_10 sf .629 .063(Lemma 7:)J f0_10 sf .126 .013( If )J f4_10 sf .187(U)A f0_10 sf .374 .037( is a minimal d-connecting path between A )J 303 654 :M -.313(and )A 320 654 :M -.17(B )A 330 654 :M .189 .019(given )J 356 654 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 368 654 :M .601 .06(in )J 380 654 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 457 654 :M -.313(and )A 474 654 :M f4_10 sf .278(E)A f0_10 sf .114 .011( )J 484 654 :M .332(is)A 59 666 :M (an edge between C and D in )S f4_10 sf (U)S f0_10 sf (, but C and D are not adjacent on )S f4_10 sf (U)S f0_10 sf (, and )S f4_10 sf <55D5>S f0_10 sf ( is the )S 383 666 :M .18 .018(result )J 409 666 :M .144 .014(of )J 421 666 :M .832 .083(substituting )J 473 666 :M f4_10 sf .278(E)A f0_10 sf .114 .011( )J 483 666 :M .222(in)A 59 678 :M .281 .028(for )J f4_10 sf .201(U)A f0_10 sf .514 .051(\(C,D\) in)J f4_10 sf .225 .022( U)J f0_10 sf .316 .032(, then either )J f4_10 sf .201(U)A f0_10 sf .313 .031(\(C,D\) is into C and )J f4_10 sf .17(E)A f0_10 sf .217 .022( is out of C, or )J f4_10 sf .201(U)A f0_10 sf .321 .032(\(C,D\) is into D, and )J f4_10 sf .17(E)A f0_10 sf .254 .025( is out of D.)J endp %%Page: 5 5 %%BeginPageSetup initializepage (peter; page: 5 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (5)S gR gS 0 0 552 730 rC 59 51 :M f2_10 sf .78(Proof.)A f0_10 sf .444 .044( )J 94 51 :M -.078(If )A 104 51 :M .755 .075(C )J 115 51 :M -.313(and )A 132 51 :M .255 .026(D )J 143 51 :M -.235(are )A 158 51 :M .515 .052(both )J 180 51 :M -.062(active )A 207 51 :M .417 .042(on )J 221 51 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(, )J 238 51 :M .202 .02(then )J 259 51 :M f4_10 sf <55D5>S f0_10 sf ( )S 273 51 :M -.181(d-connects )A 318 51 :M .255 .026(A )J 329 51 :M -.313(and )A 346 51 :M -.17(B )A 356 51 :M .189 .019(given )J 382 51 :M f2_10 sf 1.052(R)A f0_10 sf .662 .066(, )J 397 51 :M -.313(and )A 415 51 :M .694 .069(is )J 427 51 :M -.037(shorter )A 459 51 :M .202 .02(than )J 481 51 :M f4_10 sf .209(U)A f0_10 sf (,)S 59 63 :M -.02(contradicting the assumption. Hence either )A 233 63 :M .755 .075(C )J 244 63 :M .694 .069(is )J 255 63 :M .555 .056(not )J 272 63 :M -.062(active )A 299 63 :M .417 .042(on )J 313 63 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(, )J 330 63 :M .144 .014(or )J 342 63 :M .255 .026(D )J 353 63 :M .694 .069(is )J 364 63 :M .555 .056(not )J 381 63 :M -.062(active )A 408 63 :M .417 .042(on )J 422 63 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(. )J 439 63 :M .5 .05( )J 443 63 :M -.078(If )A 453 63 :M f4_10 sf <55D5>S f0_10 sf ( )S 467 63 :M -.306(agrees)A 59 75 :M .119 .012(with )J f4_10 sf .063(U)A f0_10 sf .09 .009( at C and D, then C and D are both active on )J f4_10 sf .046<55D5>A f0_10 sf .104 .01(. Hence )J f4_10 sf .088 .009<55D520>J f0_10 sf .119 .012( disagrees with )J f4_10 sf .063(U)A f0_10 sf .074 .007( at C or D.)J 59 90 :M .228 .023(If C is a collider on )J f4_10 sf .123<55D5>A f0_10 sf .212 .021(, but not on )J f4_10 sf .169(U)A f0_10 sf .254 .025(, it follows that )J f4_10 sf .143(E)A f0_10 sf .206 .021( is an edge )J 326 90 :M .255 .026(D )J 337 90 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 356 90 :M 1.108 .111(C, )J 370 90 :M -.097(there )A 393 90 :M .694 .069(is )J 404 90 :M .051 .005(an )J 417 90 :M -.344(edge )A 438 90 :M .555 .056(M )J 451 90 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 470 90 :M .755 .075(C )J 481 90 :M (on)S 59 102 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 73 102 :M -.313(and )A 90 102 :M f4_10 sf .052(U)A f0_10 sf .123 .012(\(C,D\) )J 124 102 :M .694 .069(is )J 135 102 :M .555 .056(out )J 152 102 :M .144 .014(of )J 164 102 :M 1.108 .111(C. )J 178 102 :M -.078(If )A 188 102 :M -.097(there )A 211 102 :M .694 .069(is )J 222 102 :M .417 .042(no )J 236 102 :M -.13(collider )A 269 102 :M .417 .042(on )J 283 102 :M f4_10 sf .052(U)A f0_10 sf .123 .012(\(C,D\) )J 317 102 :M .202 .02(then )J 339 102 :M .755 .075(C )J 351 102 :M .694 .069(is )J 363 102 :M -.044(either )A 390 102 :M .051 .005(an )J 404 102 :M -.101(ancestor )A 441 102 :M .144 .014(of )J 454 102 :M .65 .065(D, )J 469 102 :M .144 .014(or )J 482 102 :M -.439(an)A 59 114 :M .03 .003(ancestor of a vertex with a \322o\323 endpoint. If C is an ancestor )J 303 114 :M .144 .014(of )J 315 114 :M .056 .006(a )J 323 114 :M -.081(vertex )A 351 114 :M .517 .052(with )J 373 114 :M .056 .006(a )J 381 114 :M -.126A 398 114 :M (endpoint, )S 439 114 :M .202 .02(then )J 460 114 :M .755 .075(C )J 471 114 :M .694 .069(is )J 482 114 :M -.439(an)A 59 126 :M -.101(ancestor )A 95 126 :M .144 .014(of )J 107 126 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 118 126 :M .601 .06(in )J 130 126 :M f4_10 sf .513(G)A f0_10 sf .236<28>A f2_10 sf .553(O)A f0_10 sf .178(,)A f2_10 sf .395(S)A f0_10 sf .178(,)A f2_10 sf .474(L)A f0_10 sf .493 .049(\), )J 178 126 :M -.313(and )A 195 126 :M -.163(hence )A 221 126 :M -.097(there )A 244 126 :M -.026(cannot )A 274 126 :M .051 .005(be )J 287 126 :M .056 .006(a )J 295 126 :M .255 .026(D )J 306 126 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 325 126 :M .755 .075(C )J 336 126 :M -.344(edge )A 357 126 :M .601 .06(in )J 369 126 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\). )J 447 126 :M -.078(If )A 458 126 :M .755 .075(C )J 470 126 :M .694 .069(is )J 482 126 :M -.439(an)A 59 138 :M .13 .013(ancestor of D, this contradicts the D *)J f1_10 sf .092A f0_10 sf .104 .01( C edge. It follows that )J 318 138 :M -.097(there )A 341 138 :M .694 .069(is )J 352 138 :M .056 .006(a )J 360 138 :M -.13(collider )A 393 138 :M .417 .042(on )J 407 138 :M f4_10 sf .052(U)A f0_10 sf .123 .012(\(C,D\) )J 441 138 :M -.313(and )A 458 138 :M -.163(hence )A 484 138 :M (C)S 59 150 :M .076 .008(is an ancestor of the first collider on )J f4_10 sf (U)S f0_10 sf .109 .011(\(C,D\). )J 243 150 :M .328 .033(It )J 253 150 :M .298 .03(follows )J 287 150 :M .361 .036(that )J 306 150 :M .755 .075(C )J 317 150 :M .694 .069(is )J 328 150 :M .051 .005(an )J 341 150 :M -.101(ancestor )A 377 150 :M .144 .014(of )J 389 150 :M f2_10 sf 1.052(R)A f0_10 sf .662 .066(. )J 404 150 :M -.207(Hence )A 432 150 :M .755 .075(C )J 443 150 :M .694 .069(is )J 454 150 :M -.062(active )A 481 150 :M (on)S 59 162 :M f4_10 sf <55D5>S f0_10 sf (.)S 59 177 :M .314 .031(Similarly, if D is a collider on )J f4_10 sf .14<55D5>A f0_10 sf .232 .023( but not on )J f4_10 sf .191(U)A f0_10 sf .248 .025(, D is active on )J f4_10 sf .14<55D5>A f0_10 sf .291 .029(. It follows )J 373 177 :M .361 .036(that )J 392 177 :M -.044(either )A 418 177 :M .755 .075(C )J 429 177 :M .694 .069(is )J 440 177 :M .056 .006(a )J 448 177 :M -.13(collider )A 481 177 :M (on)S 59 189 :M f4_10 sf .169(U)A f0_10 sf .205 .021( but not on )J f4_10 sf .124<55D5>A f0_10 sf .148 .015(, or )J f4_10 sf .169(D)A f0_10 sf .221 .022( is a collider on )J f4_10 sf .169(U)A f0_10 sf .205 .021( but not on )J f4_10 sf .124<55D5>A f0_10 sf .303 .03(. Hence either )J f4_10 sf .169(U)A f0_10 sf .264 .026(\(C,D\) is into C and )J f4_10 sf .143(E)A f0_10 sf .208 .021( is out of C, or)J 59 201 :M f4_10 sf .147(U)A f0_10 sf .234 .023(\(C,D\) is into D, and )J f4_10 sf .124(E)A f0_10 sf .159 .016( is out of D. )J f1_10 sf <5C>S 59 216 :M f2_10 sf 2.491 .249(Lemma )J 100 216 :M .838(8)A f0_10 sf .804 .08(: )J 114 216 :M -.078(If )A 125 216 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 137 216 :M .694 .069(is )J 149 216 :M .056 .006(a )J 158 216 :M .688 .069(minimal )J 197 216 :M -.142(d-connecting )A 252 216 :M .202 .02(path )J 274 216 :M -.116(between )A 311 216 :M .255 .026(A )J 323 216 :M -.313(and )A 341 216 :M -.17(B )A 352 216 :M .189 .019(given )J 379 216 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 392 216 :M .601 .06(in )J 405 216 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 484 216 :M f4_10 sf (U)S 59 228 :M f0_10 sf .033 .003(contains C *)J f1_10 sf S f0_10 sf (* F *)S f1_10 sf S f0_10 sf .023 .002(* D, and C and D are adjacent )J 274 228 :M .601 .06(in )J 286 228 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 363 228 :M .202 .02(then )J 384 228 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 458 228 :M -.046(contains)A 59 240 :M .116 .012(one of the following subgraphs:)J 94 316 340 17 rC gS .625 .631 scale 148.698 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 151.896 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 155.094 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 158.292 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 161.49 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 164.687 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 167.885 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 171.083 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 172.682 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 175.88 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 179.078 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 182.275 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 185.473 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 188.671 521.233 :M f0_12 sf <28>S gR gS .625 .631 scale 191.869 521.233 :M f0_12 sf (v)S gR gS .625 .631 scale 198.264 521.233 :M f0_12 sf <29>S gR gS .625 .631 scale 203.061 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 204.66 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 207.858 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 211.056 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 214.254 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 217.451 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 220.649 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 223.847 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 227.045 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 228.644 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 231.842 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 235.039 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 238.237 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 241.435 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 244.633 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 247.831 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 251.028 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 252.627 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 255.825 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 259.023 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 262.221 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 265.419 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 268.616 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 271.814 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 275.012 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 276.611 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 279.809 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 283.007 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 286.204 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 289.402 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 292.6 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 295.798 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 298.996 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 302.193 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 303.792 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 306.99 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 310.188 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 313.386 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 316.584 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 319.781 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 322.979 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 326.177 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 327.776 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 330.974 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 334.172 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 337.369 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 340.567 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 343.765 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 346.963 521.233 :M f0_12 sf <28>S gR gS .625 .631 scale 350.161 521.233 :M f0_12 sf (v)S gR gS .625 .631 scale 356.556 521.233 :M f0_12 sf (i)S gR gS .625 .631 scale 359.754 521.233 :M f0_12 sf <29>S gR gS .625 .631 scale 362.952 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 366.15 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 369.348 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 372.545 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 375.743 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 378.941 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 382.139 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 383.738 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 386.935 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 390.133 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 393.331 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 396.529 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 399.727 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 402.925 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 406.122 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 407.721 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 410.919 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 414.117 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 417.315 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 420.513 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 423.71 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 426.908 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 430.106 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 431.705 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 434.903 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 438.101 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 441.298 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 444.496 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 447.694 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 450.892 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 454.09 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 455.688 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 458.886 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 462.084 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 465.282 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 468.48 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 471.678 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 474.875 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 478.073 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 479.672 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 482.87 521.233 :M f0_12 sf <28>S gR gS .625 .631 scale 487.667 521.233 :M f0_12 sf (v)S gR gS .625 .631 scale 494.062 521.233 :M f0_12 sf (i)S gR gS .625 .631 scale 495.661 521.233 :M f0_12 sf (i)S gR gS .625 .631 scale 498.859 521.233 :M f0_12 sf <29>S gR gS .625 .631 scale 503.656 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 506.854 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 510.051 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 511.65 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 514.848 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 518.046 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 521.244 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 524.441 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 527.639 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 530.837 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 534.035 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 535.634 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 538.832 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 542.029 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 545.227 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 548.425 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 551.623 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 554.821 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 558.019 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 559.617 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 562.815 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 566.013 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 569.211 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 572.409 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 575.607 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 578.804 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 582.002 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 583.601 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 586.799 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 589.997 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 593.194 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 596.392 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 599.59 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 602.788 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 605.986 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 609.184 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 610.782 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 613.98 521.233 :M f0_12 sf ( )S gR gS .625 .631 scale 617.178 521.233 :M f0_12 sf <28>S gR gS .625 .631 scale 621.975 521.233 :M f0_12 sf (v)S gR gS .625 .631 scale 626.772 521.233 :M f0_12 sf (i)S gR gS .625 .631 scale 629.969 521.233 :M f0_12 sf (i)S gR gS .625 .631 scale 633.167 521.233 :M f0_12 sf (i)S gR gS .625 .631 scale 636.365 521.233 :M f0_12 sf <29>S gR gR gS 106 270 340 17 rC gS .625 .631 scale 169.484 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 172.682 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 175.88 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 179.078 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 180.676 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 183.874 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 187.072 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 190.27 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 193.468 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 196.666 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 199.863 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 203.061 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 204.66 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 207.858 448.356 :M f0_12 sf <28>S gR gS .625 .631 scale 212.655 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 215.852 448.356 :M f0_12 sf <29>S gR gS .625 .631 scale 219.05 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 222.248 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 225.446 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 228.644 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 231.842 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 235.039 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 236.638 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 239.836 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 243.034 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 246.232 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 249.429 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 252.627 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 255.825 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 259.023 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 260.622 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 263.82 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 267.017 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 270.215 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 273.413 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 276.611 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 279.809 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 283.007 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 284.605 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 287.803 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 291.001 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 294.199 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 297.397 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 300.595 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 303.792 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 306.99 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 310.188 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 311.787 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 314.985 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 318.182 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 321.38 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 324.578 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 327.776 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 330.974 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 334.172 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 335.77 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 338.968 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 342.166 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 345.364 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 348.562 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 351.76 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 354.957 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 358.155 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 359.754 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 362.952 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 366.15 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 369.348 448.356 :M f0_12 sf <28>S gR gS .625 .631 scale 374.144 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 375.743 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 378.941 448.356 :M f0_12 sf <29>S gR gS .625 .631 scale 383.738 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 386.935 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 390.133 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 391.732 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 394.93 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 398.128 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 401.326 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 404.523 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 407.721 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 410.919 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 414.117 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 415.716 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 418.914 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 422.111 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 425.309 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 428.507 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 431.705 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 434.903 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 438.101 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 439.699 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 442.897 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 446.095 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 449.293 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 452.491 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 455.688 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 458.886 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 462.084 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 463.683 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 466.881 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 470.079 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 473.276 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 476.474 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 479.672 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 482.87 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 486.068 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 487.667 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 490.864 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 494.062 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 497.26 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 500.458 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 503.656 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 506.854 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 510.051 448.356 :M f0_12 sf <28>S gR gS .625 .631 scale 513.249 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 516.447 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 519.645 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 522.843 448.356 :M f0_12 sf <29>S gR gS .625 .631 scale 526.04 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 529.238 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 532.436 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 535.634 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 538.832 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 542.029 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 543.628 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 546.826 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 550.024 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 553.222 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 556.42 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 559.617 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 562.815 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 566.013 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 567.612 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 570.81 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 574.008 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 577.205 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 580.403 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 583.601 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 586.799 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 589.997 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 593.194 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 594.793 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 597.991 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 601.189 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 604.387 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 607.585 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 610.782 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 613.98 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 617.178 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 618.777 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 621.975 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 625.173 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 628.37 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 631.568 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 634.766 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 637.964 448.356 :M f0_12 sf ( )S gR gS .625 .631 scale 641.162 448.356 :M f0_12 sf <28>S gR gS .625 .631 scale 644.36 448.356 :M f0_12 sf (i)S gR gS .625 .631 scale 647.557 448.356 :M f0_12 sf (v)S gR gS .625 .631 scale 653.953 448.356 :M f0_12 sf <29>S gR gR gS 93 245 364 89 rC 131 266.75 -.75 .75 137.75 266 .75 131 266 @a np 136 264 :M 136 269 :L 141 266 :L 136 264 :L .75 lw eofill -.75 -.75 136.75 269.75 .75 .75 136 264 @b -.75 -.75 136.75 269.75 .75 .75 141 266 @b 136 264.75 -.75 .75 141.75 266 .75 136 264 @a np 132 269 :M 132 264 :L 127 266 :L 132 269 :L eofill -.75 -.75 132.75 269.75 .75 .75 132 264 @b -.75 -.75 127.75 266.75 .75 .75 132 264 @b 127 266.75 -.75 .75 132.75 269 .75 127 266 @a 151 266.75 -.75 .75 164.75 266 .75 151 266 @a np 162 264 :M 162 269 :L 167 266 :L 162 264 :L eofill -.75 -.75 162.75 269.75 .75 .75 162 264 @b -.75 -.75 162.75 269.75 .75 .75 167 266 @b 162 264.75 -.75 .75 167.75 266 .75 162 264 @a 96 256 81 19 rC gS .625 .631 scale 153.495 426.176 :M f0_12 sf (A)S gR gS .625 .631 scale 161.49 426.176 :M f0_12 sf (*)S gR gS .625 .631 scale 167.885 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 171.083 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 172.682 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 175.88 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 179.078 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 182.275 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 185.473 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 188.671 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 191.869 426.176 :M f0_12 sf (C)S gR gS .625 .631 scale 199.863 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 203.061 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 204.66 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 207.858 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 211.056 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 214.254 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 217.451 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 220.649 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 223.847 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 227.045 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 228.644 426.176 :M f0_12 sf (F)S gR gS .625 .631 scale 236.638 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 239.836 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 243.034 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 244.633 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 247.831 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 251.028 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 254.226 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 257.424 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 260.622 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 263.82 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 267.017 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 268.616 426.176 :M f0_12 sf (D)S gR gR gS 93 245 364 89 rC -.75 -.75 174.75 260.75 .75 .75 174 255 @b np 177 258 :M 172 258 :L 174 263 :L 177 258 :L .75 lw eofill 172 258.75 -.75 .75 177.75 258 .75 172 258 @a 172 258.75 -.75 .75 174.75 263 .75 172 258 @a -.75 -.75 174.75 263.75 .75 .75 177 258 @b -180 -90 45 28 147 260.5 @n -90 0 55 21 147 257 @n 104 266.75 -.75 .75 114.75 266 .75 104 266 @a np 113 264 :M 113 269 :L 118 266 :L 113 264 :L eofill -.75 -.75 113.75 269.75 .75 .75 113 264 @b -.75 -.75 113.75 269.75 .75 .75 118 266 @b 113 264.75 -.75 .75 118.75 266 .75 113 264 @a 223 266.75 -.75 .75 229.75 266 .75 223 266 @a np 228 264 :M 228 269 :L 233 266 :L 228 264 :L eofill -.75 -.75 228.75 269.75 .75 .75 228 264 @b -.75 -.75 228.75 269.75 .75 .75 233 266 @b 228 264.75 -.75 .75 233.75 266 .75 228 264 @a np 224 269 :M 224 264 :L 219 266 :L 224 269 :L eofill -.75 -.75 224.75 269.75 .75 .75 224 264 @b -.75 -.75 219.75 266.75 .75 .75 224 264 @b 219 266.75 -.75 .75 224.75 269 .75 219 266 @a 186 257 81 19 rC gS .625 .631 scale 295.798 427.76 :M f0_12 sf (A)S gR gS .625 .631 scale 303.792 427.76 :M f0_12 sf (*)S gR gS .625 .631 scale 310.188 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 313.386 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 316.584 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 319.781 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 322.979 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 326.177 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 327.776 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 330.974 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 334.172 427.76 :M f0_12 sf (C)S gR gS .625 .631 scale 342.166 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 345.364 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 348.562 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 351.76 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 354.957 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 358.155 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 359.754 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 362.952 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 366.15 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 369.348 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 372.545 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 375.743 427.76 :M f0_12 sf (F)S gR gS .625 .631 scale 382.139 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 385.337 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 388.534 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 391.732 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 394.93 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 398.128 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 399.727 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 402.925 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 406.122 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 409.32 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 412.518 427.76 :M f0_12 sf ( )S gR gS .625 .631 scale 415.716 427.76 :M f0_12 sf (D)S gR gR gS 93 245 364 89 rC -.75 -.75 262.75 260.75 .75 .75 262 255 @b np 264 258 :M 259 258 :L 262 263 :L 264 258 :L .75 lw eofill 259 258.75 -.75 .75 264.75 258 .75 259 258 @a 259 258.75 -.75 .75 262.75 263 .75 259 258 @a -.75 -.75 262.75 263.75 .75 .75 264 258 @b -180 -90 44 28 234.5 260.5 @n -90 0 54 21 234.5 257 @n 194 267.75 -.75 .75 204.75 267 .75 194 267 @a np 202 265 :M 202 270 :L 207 267 :L 202 265 :L eofill -.75 -.75 202.75 270.75 .75 .75 202 265 @b -.75 -.75 202.75 270.75 .75 .75 207 267 @b 202 265.75 -.75 .75 207.75 267 .75 202 265 @a 328 266.75 -.75 .75 340.75 266 .75 328 266 @a np 339 264 :M 339 269 :L 344 266 :L 339 264 :L eofill -.75 -.75 339.75 269.75 .75 .75 339 264 @b -.75 -.75 339.75 269.75 .75 .75 344 266 @b 339 264.75 -.75 .75 344.75 266 .75 339 264 @a 272 256 81 19 rC gS .625 .631 scale 434.903 426.176 :M f0_12 sf (A)S gR gS .625 .631 scale 442.897 426.176 :M f0_12 sf (*)S gR gS .625 .631 scale 447.694 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 450.892 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 454.09 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 457.287 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 460.485 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 463.683 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 466.881 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 470.079 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 471.678 426.176 :M f0_12 sf (C)S gR gS .625 .631 scale 479.672 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 482.87 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 486.068 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 489.266 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 492.463 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 495.661 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 498.859 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 502.057 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 503.656 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 506.854 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 510.051 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 513.249 426.176 :M f0_12 sf (F)S gR gS .625 .631 scale 519.645 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 522.843 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 526.04 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 529.238 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 532.436 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 535.634 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 538.832 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 542.029 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 543.628 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 546.826 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 550.024 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 553.222 426.176 :M f0_12 sf (D)S gR gR gS 93 245 364 89 rC -.75 -.75 347.75 259.75 .75 .75 347 254 @b np 349 258 :M 344 258 :L 347 263 :L 349 258 :L .75 lw eofill 344 258.75 -.75 .75 349.75 258 .75 344 258 @a 344 258.75 -.75 .75 347.75 263 .75 344 258 @a -.75 -.75 347.75 263.75 .75 .75 349 258 @b -180 -90 45 29 320 260 @n -90 0 55 21 320 256 @n 281 266.75 -.75 .75 291.75 266 .75 281 266 @a np 289 263 :M 289 268 :L 294 266 :L 289 263 :L eofill -.75 -.75 289.75 268.75 .75 .75 289 263 @b -.75 -.75 289.75 268.75 .75 .75 294 266 @b 289 263.75 -.75 .75 294.75 266 .75 289 263 @a 355 256 88 19 rC gS .625 .631 scale 567.612 426.176 :M f0_12 sf (A)S gR gS .625 .631 scale 575.607 426.176 :M f0_12 sf (*)S gR gS .625 .631 scale 582.002 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 583.601 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 586.799 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 589.997 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 593.194 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 596.392 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 599.59 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 602.788 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 605.986 426.176 :M f0_12 sf (C)S gR gS .625 .631 scale 613.98 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 617.178 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 618.777 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 621.975 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 625.173 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 628.37 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 631.568 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 634.766 426.176 :M f0_12 sf (o)S gR gS .625 .631 scale 641.162 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 642.761 426.176 :M f0_12 sf (F)S gR gS .625 .631 scale 650.755 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 653.953 426.176 :M f0_12 sf (o)S gR gS .625 .631 scale 658.75 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 661.947 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 665.145 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 668.343 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 671.541 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 674.739 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 677.937 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 681.134 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 682.733 426.176 :M f0_12 sf ( )S gR gS .625 .631 scale 685.931 426.176 :M f0_12 sf (D)S gR gR gS 93 245 364 89 rC -.75 -.75 432.75 259.75 .75 .75 432 254 @b np 434 258 :M 429 258 :L 432 263 :L 434 258 :L .75 lw eofill 429 258.75 -.75 .75 434.75 258 .75 429 258 @a 429 258.75 -.75 .75 432.75 263 .75 429 258 @a -.75 -.75 432.75 263.75 .75 .75 434 258 @b -180 -90 45 29 405 260 @n -90 0 55 21 405 256 @n 364 266.75 -.75 .75 374.75 266 .75 364 266 @a np 373 264 :M 373 269 :L 378 266 :L 373 264 :L eofill -.75 -.75 373.75 269.75 .75 .75 373 264 @b -.75 -.75 373.75 269.75 .75 .75 378 266 @b 373 264.75 -.75 .75 378.75 266 .75 373 264 @a 245 266.75 -.75 .75 251.75 266 .75 245 266 @a np 250 264 :M 250 269 :L 255 266 :L 250 264 :L eofill -.75 -.75 250.75 269.75 .75 .75 250 264 @b -.75 -.75 250.75 269.75 .75 .75 255 266 @b 250 264.75 -.75 .75 255.75 266 .75 250 264 @a np 246 269 :M 246 264 :L 241 266 :L 246 269 :L eofill -.75 -.75 246.75 269.75 .75 .75 246 264 @b -.75 -.75 241.75 266.75 .75 .75 246 264 @b 241 266.75 -.75 .75 246.75 269 .75 241 266 @a 306 266.75 -.75 .75 318.75 266 .75 306 266 @a np 307 269 :M 307 264 :L 302 266 :L 307 269 :L eofill -.75 -.75 307.75 269.75 .75 .75 307 264 @b -.75 -.75 302.75 266.75 .75 .75 307 264 @b 302 266.75 -.75 .75 307.75 269 .75 302 266 @a 389 266.75 -.75 .75 397.75 266 .75 389 266 @a np 391 269 :M 391 264 :L 386 266 :L 391 269 :L eofill -.75 -.75 391.75 269.75 .75 .75 391 264 @b -.75 -.75 386.75 266.75 .75 .75 391 264 @b 386 266.75 -.75 .75 391.75 269 .75 386 266 @a 413 266.75 -.75 .75 421.75 266 .75 413 266 @a np 419 264 :M 419 269 :L 424 266 :L 419 264 :L eofill -.75 -.75 419.75 269.75 .75 .75 419 264 @b -.75 -.75 419.75 269.75 .75 .75 424 266 @b 419 264.75 -.75 .75 424.75 266 .75 419 264 @a 134 312.75 -.75 .75 140.75 312 .75 134 312 @a np 139 309 :M 139 314 :L 144 312 :L 139 309 :L eofill -.75 -.75 139.75 314.75 .75 .75 139 309 @b -.75 -.75 139.75 314.75 .75 .75 144 312 @b 139 309.75 -.75 .75 144.75 312 .75 139 309 @a np 135 314 :M 135 309 :L 130 312 :L 135 314 :L eofill -.75 -.75 135.75 314.75 .75 .75 135 309 @b -.75 -.75 130.75 312.75 .75 .75 135 309 @b 130 312.75 -.75 .75 135.75 314 .75 130 312 @a 105 312.75 -.75 .75 117.75 312 .75 105 312 @a np 106 314 :M 106 309 :L 101 312 :L 106 314 :L eofill -.75 -.75 106.75 314.75 .75 .75 106 309 @b -.75 -.75 101.75 312.75 .75 .75 106 309 @b 101 312.75 -.75 .75 106.75 314 .75 101 312 @a 368 304 88 19 rC gS .625 .631 scale 586.799 502.222 :M f0_12 sf (C)S gR gS .625 .631 scale 594.793 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 597.991 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 601.189 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 604.387 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 607.585 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 610.782 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 613.98 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 617.178 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 618.777 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 621.975 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 625.173 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 628.37 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 631.568 502.222 :M f0_12 sf (F)S gR gS .625 .631 scale 637.964 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 641.162 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 644.36 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 647.557 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 650.755 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 653.953 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 657.151 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 658.75 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 661.947 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 665.145 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 668.343 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 671.541 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 674.739 502.222 :M f0_12 sf (D)S gR gS .625 .631 scale 682.733 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 685.931 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 689.129 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 690.728 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 693.926 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 697.123 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 700.321 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 703.519 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 706.717 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 709.915 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 713.113 502.222 :M f0_12 sf (*)S gR gS .625 .631 scale 717.909 502.222 :M f0_12 sf (E)S gR gR gS 277 304 89 19 rC gS .625 .631 scale 442.897 502.222 :M f0_12 sf (C)S gR gS .625 .631 scale 450.892 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 454.09 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 455.688 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 458.886 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 462.084 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 465.282 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 468.48 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 471.678 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 474.875 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 478.073 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 479.672 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 482.87 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 486.068 502.222 :M f0_12 sf (F)S gR gS .625 .631 scale 494.062 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 495.661 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 498.859 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 502.057 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 505.255 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 508.452 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 511.65 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 514.848 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 518.046 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 519.645 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 522.843 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 526.04 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 529.238 502.222 :M f0_12 sf (D)S gR gS .625 .631 scale 537.233 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 540.431 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 543.628 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 546.826 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 550.024 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 551.623 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 554.821 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 558.019 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 561.216 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 564.414 502.222 :M f0_12 sf ( )S gR gS .625 .631 scale 567.612 502.222 :M f0_12 sf (*)S gR gS .625 .631 scale 574.008 502.222 :M f0_12 sf (E)S gR gR gS 186 302 89 19 rC gS .625 .631 scale 297.397 499.053 :M f0_12 sf (C)S gR gS .625 .631 scale 305.391 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 308.589 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 311.787 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 314.985 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 318.182 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 319.781 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 322.979 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 326.177 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 329.375 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 332.573 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 335.77 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 338.968 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 342.166 499.053 :M f0_12 sf (F)S gR gS .625 .631 scale 348.562 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 351.76 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 354.957 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 358.155 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 359.754 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 362.952 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 366.15 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 369.348 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 372.545 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 375.743 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 378.941 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 382.139 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 383.738 499.053 :M f0_12 sf (D)S gR gS .625 .631 scale 391.732 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 394.93 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 398.128 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 401.326 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 404.523 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 407.721 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 410.919 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 414.117 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 415.716 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 418.914 499.053 :M f0_12 sf ( )S gR gS .625 .631 scale 422.111 499.053 :M f0_12 sf (*)S gR gS .625 .631 scale 428.507 499.053 :M f0_12 sf (E)S gR gR gS 94 302 88 19 rC gS .625 .631 scale 148.698 500.638 :M f0_12 sf (C)S gR gS .625 .631 scale 156.693 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 159.891 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 163.089 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 166.286 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 169.484 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 172.682 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 175.88 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 179.078 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 180.676 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 183.874 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 187.072 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 190.27 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 193.468 500.638 :M f0_12 sf (F)S gR gS .625 .631 scale 199.863 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 203.061 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 206.259 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 209.457 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 212.655 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 215.852 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 219.05 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 220.649 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 223.847 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 227.045 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 230.243 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 233.44 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 236.638 500.638 :M f0_12 sf (D)S gR gS .625 .631 scale 244.633 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 247.831 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 251.028 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 252.627 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 255.825 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 259.023 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 262.221 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 265.419 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 268.616 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 271.814 500.638 :M f0_12 sf ( )S gR gS .625 .631 scale 275.012 500.638 :M f0_12 sf (*)S gR gS .625 .631 scale 279.809 500.638 :M f0_12 sf (E)S gR gR gS 93 245 364 89 rC -.75 -.75 99.75 305.75 .75 .75 99 301 @b np 102 304 :M 97 304 :L 99 309 :L 102 304 :L .75 lw eofill 97 304.75 -.75 .75 102.75 304 .75 97 304 @a 97 304.75 -.75 .75 99.75 309 .75 97 304 @a -.75 -.75 99.75 309.75 .75 .75 102 304 @b -90 0 43 29 128 306 @n -180 -90 56 21 127.5 302 @n 162 312.75 -.75 .75 172.75 312 .75 162 312 @a np 163 314 :M 163 309 :L 158 312 :L 163 314 :L eofill -.75 -.75 163.75 314.75 .75 .75 163 309 @b -.75 -.75 158.75 312.75 .75 .75 163 309 @b 158 312.75 -.75 .75 163.75 314 .75 158 312 @a 201 312.75 -.75 .75 207.75 312 .75 201 312 @a np 206 309 :M 206 314 :L 211 312 :L 206 309 :L eofill -.75 -.75 206.75 314.75 .75 .75 206 309 @b -.75 -.75 206.75 314.75 .75 .75 211 312 @b 206 309.75 -.75 .75 211.75 312 .75 206 309 @a np 202 314 :M 202 309 :L 197 312 :L 202 314 :L eofill -.75 -.75 202.75 314.75 .75 .75 202 309 @b -.75 -.75 197.75 312.75 .75 .75 202 309 @b 197 312.75 -.75 .75 202.75 314 .75 197 312 @a 310 313.75 -.75 .75 323.75 313 .75 310 313 @a np 321 311 :M 321 316 :L 326 313 :L 321 311 :L eofill -.75 -.75 321.75 316.75 .75 .75 321 311 @b -.75 -.75 321.75 316.75 .75 .75 326 313 @b 321 311.75 -.75 .75 326.75 313 .75 321 311 @a 223 312.75 -.75 .75 229.75 312 .75 223 312 @a np 228 309 :M 228 314 :L 233 312 :L 228 309 :L eofill -.75 -.75 228.75 314.75 .75 .75 228 309 @b -.75 -.75 228.75 314.75 .75 .75 233 312 @b 228 309.75 -.75 .75 233.75 312 .75 228 309 @a np 224 314 :M 224 309 :L 219 312 :L 224 314 :L eofill -.75 -.75 224.75 314.75 .75 .75 224 309 @b -.75 -.75 219.75 312.75 .75 .75 224 309 @b 219 312.75 -.75 .75 224.75 314 .75 219 312 @a 288 313.75 -.75 .75 301.75 313 .75 288 313 @a np 289 316 :M 289 311 :L 284 313 :L 289 316 :L eofill -.75 -.75 289.75 316.75 .75 .75 289 311 @b -.75 -.75 284.75 313.75 .75 .75 289 311 @b 284 313.75 -.75 .75 289.75 316 .75 284 313 @a 379 312.75 -.75 .75 387.75 312 .75 379 312 @a np 381 314 :M 381 309 :L 376 312 :L 381 314 :L eofill -.75 -.75 381.75 314.75 .75 .75 381 309 @b -.75 -.75 376.75 312.75 .75 .75 381 309 @b 376 312.75 -.75 .75 381.75 314 .75 376 312 @a 403 312.75 -.75 .75 411.75 312 .75 403 312 @a np 409 309 :M 409 314 :L 414 312 :L 409 309 :L eofill -.75 -.75 409.75 314.75 .75 .75 409 309 @b -.75 -.75 409.75 314.75 .75 .75 414 312 @b 409 309.75 -.75 .75 414.75 312 .75 409 309 @a -.75 -.75 191.75 305.75 .75 .75 191 301 @b np 193 304 :M 188 304 :L 191 309 :L 193 304 :L eofill 188 304.75 -.75 .75 193.75 304 .75 188 304 @a 188 304.75 -.75 .75 191.75 309 .75 188 304 @a -.75 -.75 191.75 309.75 .75 .75 193 304 @b -90 0 43 29 219 306 @n -180 -90 55 21 219 302 @n -.75 -.75 280.75 306.75 .75 .75 280 302 @b np 283 305 :M 278 305 :L 280 310 :L 283 305 :L eofill 278 305.75 -.75 .75 283.75 305 .75 278 305 @a 278 305.75 -.75 .75 280.75 310 .75 278 305 @a -.75 -.75 280.75 310.75 .75 .75 283 305 @b -90 0 44 29 308.5 307 @n -180 -90 56 21 308.5 303 @n -.75 -.75 371.75 305.75 .75 .75 371 301 @b np 373 304 :M 368 304 :L 371 309 :L 373 304 :L eofill 368 304.75 -.75 .75 373.75 304 .75 368 304 @a 368 304.75 -.75 .75 371.75 309 .75 368 304 @a -.75 -.75 371.75 309.75 .75 .75 373 304 @b -90 0 43 29 399 306 @n -180 -90 56 21 399.5 302 @n 254 313.75 -.75 .75 264.75 313 .75 254 313 @a np 255 315 :M 255 310 :L 250 313 :L 255 315 :L eofill -.75 -.75 255.75 315.75 .75 .75 255 310 @b -.75 -.75 250.75 313.75 .75 .75 255 310 @b 250 313.75 -.75 .75 255.75 315 .75 250 313 @a 344 314.75 -.75 .75 354.75 314 .75 344 314 @a np 345 316 :M 345 311 :L 340 314 :L 345 316 :L eofill -.75 -.75 345.75 316.75 .75 .75 345 311 @b -.75 -.75 340.75 314.75 .75 .75 345 311 @b 340 314.75 -.75 .75 345.75 316 .75 340 314 @a 432 313.75 -.75 .75 442.75 313 .75 432 313 @a np 433 316 :M 433 311 :L 428 313 :L 433 316 :L eofill -.75 -.75 433.75 316.75 .75 .75 433 311 @b -.75 -.75 428.75 313.75 .75 .75 433 311 @b 428 313.75 -.75 .75 433.75 316 .75 428 313 @a gR gS 0 0 552 730 rC 254 347 :M f2_10 sf 3.634 .363(Figure 1)J 59 362 :M .78(Proof.)A f0_10 sf .444 .044( )J 94 362 :M .134 .013(Let )J 111 362 :M f4_10 sf .278(E)A f0_10 sf .114 .011( )J 121 362 :M .051 .005(be )J 134 362 :M .218 .022(the )J 150 362 :M -.344(edge )A 171 362 :M -.116(between )A 207 362 :M .755 .075(C )J 218 362 :M -.313(and )A 235 362 :M .65 .065(D, )J 249 362 :M -.313(and )A 266 362 :M f4_10 sf <55D5>S f0_10 sf ( )S 280 362 :M .051 .005(be )J 293 362 :M .218 .022(the )J 309 362 :M .18 .018(result )J 336 362 :M .144 .014(of )J 349 362 :M .832 .083(substituting )J 402 362 :M f4_10 sf .278(E)A f0_10 sf .114 .011( )J 413 362 :M .601 .06(in )J 426 362 :M -.052(for )A 442 362 :M f4_10 sf .151(U)A f0_10 sf .366 .037(\(C,D\). )J 480 362 :M -.67(By)A 59 374 :M .445 .045(Lemma 7, either )J f4_10 sf .216(U)A f0_10 sf .338 .034(\(C,D\) is into C and )J f4_10 sf .183(E)A f0_10 sf .235 .023( is out of C, or )J f4_10 sf .216(U)A f0_10 sf .346 .035(\(C,D\) is into D, and )J f4_10 sf .183(E)A f0_10 sf .238 .024( is out of D. )J 437 374 :M .361 .036(Suppose )J 475 374 :M -.025(first)A 59 386 :M .361 .036(that )J 78 386 :M .356 .036(E )J 88 386 :M .694 .069(is )J 99 386 :M .555 .056(out )J 116 386 :M .144 .014(of )J 128 386 :M .755 .075(C )J 139 386 :M -.313(and )A 156 386 :M f4_10 sf .052(U)A f0_10 sf .123 .012(\(C,D\) )J 190 386 :M .694 .069(is )J 201 386 :M .674 .067(into )J 221 386 :M 1.108 .111(C. )J 235 386 :M (Then )S 259 386 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 333 386 :M .098 .01(contains )J 370 386 :M -.044(either )A 396 386 :M .405 .04(\(i\), )J 412 386 :M .525 .053(\(ii\), )J 431 386 :M .34 .034(\(iii\) )J 451 386 :M .144 .014(or )J 464 386 :M .047 .005(\(iv\) )J 483 386 :M -.328(of)A 59 398 :M .192 .019(Figure 1 or one of the following subgraphs \(ix\), \(x\), or \(xi\) of Figure 2:)J 322 417 78 19 rC gS .634 .632 scale 507.823 680.014 :M f0_12 sf (A)S gR gS .634 .632 scale 515.709 680.014 :M f0_12 sf (*)S gR gS .634 .632 scale 520.44 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 523.594 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 526.748 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 529.902 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 533.057 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 536.211 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 539.365 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 542.519 680.014 :M f0_12 sf (C)S gR gS .634 .632 scale 550.405 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 553.559 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 556.713 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 559.867 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 561.444 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 564.598 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 567.753 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 570.907 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 574.061 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 577.215 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 580.369 680.014 :M f0_12 sf (F)S gR gS .634 .632 scale 586.678 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 589.832 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 592.986 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 596.14 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 599.294 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 602.449 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 605.603 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 608.757 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 610.334 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 613.488 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 616.642 680.014 :M f0_12 sf ( )S gR gS .634 .632 scale 619.797 680.014 :M f0_12 sf (D)S gR gR gS 237 415 78 19 rC gS .634 .632 scale 373.77 676.852 :M f0_12 sf (A)S gR gS .634 .632 scale 381.656 676.852 :M f0_12 sf (*)S gR gS .634 .632 scale 386.387 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 389.541 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 392.696 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 395.85 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 399.004 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 402.158 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 405.312 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 408.466 676.852 :M f0_12 sf (C)S gR gS .634 .632 scale 416.352 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 419.506 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 422.66 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 425.814 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 427.392 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 430.546 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 433.7 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 436.854 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 440.008 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 443.162 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 446.317 676.852 :M f0_12 sf (F)S gR gS .634 .632 scale 452.625 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 455.779 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 458.933 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 462.088 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 465.242 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 468.396 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 471.55 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 474.704 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 477.858 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 479.436 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 482.59 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 485.744 676.852 :M f0_12 sf (D)S gR gR gS 149 415 78 19 rC gS .634 .632 scale 233.409 676.852 :M f0_12 sf (A)S gR gS .634 .632 scale 242.872 676.852 :M f0_12 sf (*)S gR gS .634 .632 scale 247.603 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 250.757 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 253.912 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 257.066 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 260.22 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 263.374 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 266.528 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 269.682 676.852 :M f0_12 sf (C)S gR gS .634 .632 scale 277.568 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 280.722 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 283.876 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 285.453 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 288.608 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 291.762 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 294.916 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 298.07 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 301.224 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 304.378 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 307.533 676.852 :M f0_12 sf (F)S gR gS .634 .632 scale 313.841 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 316.995 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 320.149 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 323.304 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 326.458 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 329.612 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 332.766 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 334.343 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 337.497 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 340.652 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 343.806 676.852 :M f0_12 sf ( )S gR gS .634 .632 scale 346.96 676.852 :M f0_12 sf (D)S gR gR gS 148 403 253 43 rC 180 425.75 -.75 .75 193.75 425 .75 180 425 @a np 182 428 :M 182 423 :L 177 425 :L 182 428 :L .75 lw eofill -.75 -.75 182.75 428.75 .75 .75 182 423 @b -.75 -.75 177.75 425.75 .75 .75 182 423 @b 177 425.75 -.75 .75 182.75 428 .75 177 425 @a 204 425.75 -.75 .75 216.75 425 .75 204 425 @a np 205 428 :M 205 423 :L 200 425 :L 205 428 :L eofill -.75 -.75 205.75 428.75 .75 .75 205 423 @b -.75 -.75 200.75 425.75 .75 .75 205 423 @b 200 425.75 -.75 .75 205.75 428 .75 200 425 @a -.75 -.75 222.75 417.75 .75 .75 222 412 @b np 225 416 :M 220 416 :L 222 421 :L 225 416 :L eofill 220 416.75 -.75 .75 225.75 416 .75 220 416 @a 220 416.75 -.75 .75 222.75 421 .75 220 416 @a -.75 -.75 222.75 421.75 .75 .75 225 416 @b -180 -90 46 29 194.5 418 @n -90 0 56 21 194.5 414 @n 268 425.75 -.75 .75 277.75 425 .75 268 425 @a np 276 423 :M 276 428 :L 281 425 :L 276 423 :L eofill -.75 -.75 276.75 428.75 .75 .75 276 423 @b -.75 -.75 276.75 428.75 .75 .75 281 425 @b 276 423.75 -.75 .75 281.75 425 .75 276 423 @a np 270 428 :M 270 423 :L 265 425 :L 270 428 :L eofill -.75 -.75 270.75 428.75 .75 .75 270 423 @b -.75 -.75 265.75 425.75 .75 .75 270 423 @b 265 425.75 -.75 .75 270.75 428 .75 265 425 @a 292 425.75 -.75 .75 305.75 425 .75 292 425 @a np 293 428 :M 293 423 :L 288 425 :L 293 428 :L eofill -.75 -.75 293.75 428.75 .75 .75 293 423 @b -.75 -.75 288.75 425.75 .75 .75 293 423 @b 288 425.75 -.75 .75 293.75 428 .75 288 425 @a -.75 -.75 310.75 417.75 .75 .75 310 412 @b np 313 416 :M 308 416 :L 310 421 :L 313 416 :L eofill 308 416.75 -.75 .75 313.75 416 .75 308 416 @a 308 416.75 -.75 .75 310.75 421 .75 308 416 @a -.75 -.75 310.75 421.75 .75 .75 313 416 @b -180 -90 45 29 283 418 @n -90 0 55 21 283 414 @n 353 426.75 -.75 .75 366.75 426 .75 353 426 @a np 355 429 :M 355 424 :L 350 426 :L 355 429 :L eofill -.75 -.75 355.75 429.75 .75 .75 355 424 @b -.75 -.75 350.75 426.75 .75 .75 355 424 @b 350 426.75 -.75 .75 355.75 429 .75 350 426 @a 377 426.75 -.75 .75 386.75 426 .75 377 426 @a np 385 424 :M 385 429 :L 390 426 :L 385 424 :L eofill -.75 -.75 385.75 429.75 .75 .75 385 424 @b -.75 -.75 385.75 429.75 .75 .75 390 426 @b 385 424.75 -.75 .75 390.75 426 .75 385 424 @a np 378 429 :M 378 424 :L 373 426 :L 378 429 :L eofill -.75 -.75 378.75 429.75 .75 .75 378 424 @b -.75 -.75 373.75 426.75 .75 .75 378 424 @b 373 426.75 -.75 .75 378.75 429 .75 373 426 @a -.75 -.75 395.75 418.75 .75 .75 395 414 @b np 398 417 :M 393 417 :L 395 422 :L 398 417 :L eofill 393 417.75 -.75 .75 398.75 417 .75 393 417 @a 393 417.75 -.75 .75 395.75 422 .75 393 417 @a -.75 -.75 395.75 422.75 .75 .75 398 417 @b -180 -90 45 29 368 419 @n -90 0 56 21 367.5 415 @n 171 428 222 17 rC gS .634 .632 scale 269.682 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 272.837 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 275.991 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 279.145 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 282.299 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 285.453 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 288.608 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 291.762 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 293.339 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 296.493 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 299.647 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 302.801 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 305.956 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 309.11 697.41 :M f0_12 sf <28>S gR gS .634 .632 scale 313.841 697.41 :M f0_12 sf (i)S gR gS .634 .632 scale 315.418 697.41 :M f0_12 sf (x)S gR gS .634 .632 scale 321.726 697.41 :M f0_12 sf <29>S gR gS .634 .632 scale 326.458 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 329.612 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 332.766 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 334.343 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 337.497 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 340.652 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 343.806 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 346.96 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 350.114 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 353.268 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 356.422 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 359.577 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 362.731 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 365.885 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 367.462 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 370.616 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 373.77 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 376.925 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 380.079 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 383.233 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 386.387 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 389.541 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 392.696 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 395.85 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 397.427 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 400.581 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 403.735 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 406.889 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 410.044 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 413.198 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 416.352 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 419.506 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 422.66 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 425.814 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 427.392 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 430.546 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 433.7 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 436.854 697.41 :M f0_12 sf <28>S gR gS .634 .632 scale 441.585 697.41 :M f0_12 sf (x)S gR gS .634 .632 scale 447.894 697.41 :M f0_12 sf <29>S gR gS .634 .632 scale 451.048 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 454.202 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 457.356 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 460.51 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 463.665 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 466.819 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 468.396 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 471.55 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 474.704 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 477.858 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 481.013 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 484.167 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 487.321 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 490.475 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 493.629 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 496.784 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 498.361 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 501.515 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 504.669 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 507.823 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 510.977 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 514.132 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 517.286 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 520.44 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 523.594 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 526.748 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 529.902 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 531.48 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 534.634 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 537.788 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 540.942 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 544.096 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 547.25 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 550.405 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 553.559 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 556.713 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 559.867 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 561.444 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 564.598 697.41 :M f0_12 sf ( )S gR gS .634 .632 scale 567.753 697.41 :M f0_12 sf <28>S gR gS .634 .632 scale 572.484 697.41 :M f0_12 sf (x)S gR gS .634 .632 scale 578.792 697.41 :M f0_12 sf (i)S gR gS .634 .632 scale 580.369 697.41 :M f0_12 sf <29>S gR gR gS 148 403 253 43 rC 156 425.75 -.75 .75 164.75 425 .75 156 425 @a np 163 423 :M 163 428 :L 168 425 :L 163 423 :L .75 lw eofill -.75 -.75 163.75 428.75 .75 .75 163 423 @b -.75 -.75 163.75 428.75 .75 .75 168 425 @b 163 423.75 -.75 .75 168.75 425 .75 163 423 @a 245 425.75 -.75 .75 253.75 425 .75 245 425 @a np 252 423 :M 252 428 :L 257 425 :L 252 423 :L eofill -.75 -.75 252.75 428.75 .75 .75 252 423 @b -.75 -.75 252.75 428.75 .75 .75 257 425 @b 252 423.75 -.75 .75 257.75 425 .75 252 423 @a 330 427.75 -.75 .75 338.75 427 .75 330 427 @a np 337 425 :M 337 430 :L 342 427 :L 337 425 :L eofill -.75 -.75 337.75 430.75 .75 .75 337 425 @b -.75 -.75 337.75 430.75 .75 .75 342 427 @b 337 425.75 -.75 .75 342.75 427 .75 337 425 @a gR gS 0 0 552 730 rC 254 459 :M f2_10 sf 3.634 .363(Figure 2)J 59 474 :M f0_10 sf -.163(However )A 98 474 :M .047 .005(\(ix\) )J 116 474 :M .098 .01(contains )J 153 474 :M .056 .006(a )J 161 474 :M -.016(cycle. )A 188 474 :M -.052(\(x\) )A 203 474 :M .694 .069(is )J 214 474 :M .585 .058(impossible )J 262 474 :M -.163(because )A 296 474 :M .755 .075(C )J 307 474 :M f1_10 sf .065A f0_10 sf ( )S 321 474 :M .855 .086(F )J 331 474 :M .623 .062(implies )J 366 474 :M .755 .075(C )J 378 474 :M .694 .069(is )J 390 474 :M .555 .056(not )J 408 474 :M .051 .005(an )J 422 474 :M -.101(ancestor )A 459 474 :M .144 .014(of )J 472 474 :M .855 .086(F )J 483 474 :M .222(in)A 59 486 :M f4_10 sf .218(G)A f0_10 sf .101<28>A f2_10 sf .235(O)A f0_10 sf .076(,)A f2_10 sf .168(S)A f0_10 sf .076(,)A f2_10 sf .202(L)A f0_10 sf .29 .029(\), but there is a path C )J f1_10 sf .298A f0_10 sf .161 .016( D )J f1_10 sf .298A f0_10 sf .359 .036( F in MAG\()J f4_10 sf .218(G)A f0_10 sf .101<28>A f2_10 sf .235(O)A f0_10 sf .076(,)A f2_10 sf .168(S)A f0_10 sf .076(,)A f2_10 sf .202(L)A f0_10 sf .383 .038(\)\), and hence )J 365 486 :M .056 .006(a )J 373 486 :M -.337(directed )A 406 486 :M .202 .02(path )J 427 486 :M .047 .005(from )J 450 486 :M .755 .075(C )J 461 486 :M .601 .06(to )J 473 486 :M .855 .086(F )J 483 486 :M .222(in)A 59 498 :M f4_10 sf .225(G)A f0_10 sf .104<28>A f2_10 sf .243(O)A f0_10 sf .078(,)A f2_10 sf .174(S)A f0_10 sf .078(,)A f2_10 sf .208(L)A f0_10 sf .415 .042(\). \(xi\) is impossible because F )J f1_10 sf .325A f0_10 sf .323 .032( D implies F is not an ancestor of D in )J f4_10 sf .225(G)A f0_10 sf .104<28>A f2_10 sf .243(O)A f0_10 sf .078(,)A f2_10 sf .174(S)A f0_10 sf .078(,)A f2_10 sf .208(L)A f0_10 sf .333 .033(\), but there is a)J 59 510 :M .219 .022(path F )J f1_10 sf .21A f0_10 sf .108 .011( C )J f1_10 sf .21A f0_10 sf .262 .026( D in MAG\()J f4_10 sf .154(G)A f0_10 sf .071<28>A f2_10 sf .166(O)A f0_10 sf .053(,)A f2_10 sf .118(S)A f0_10 sf .053(,)A f2_10 sf .142(L)A f0_10 sf .242 .024(\)\), and hence a directed path from F to D in )J f4_10 sf .154(G)A f0_10 sf .071<28>A f2_10 sf .166(O)A f0_10 sf .053(,)A f2_10 sf .118(S)A f0_10 sf .053(,)A f2_10 sf .142(L)A f0_10 sf .124(\).)A 59 525 :M .198 .02(Similarly, it can be shown that if there is an edge D )J f1_10 sf .163A f0_10 sf ( )S 284 525 :M 1.108 .111(C, )J 298 525 :M .218 .022(the )J 314 525 :M .515 .052(only )J 336 525 :M .404 .04(possible )J 373 525 :M (subgraphs )S 417 525 :M -.235(are )A 432 525 :M .246 .025(\(v\), )J 450 525 :M .378 .038(\(vi\), )J 471 525 :M .058(\(vii\),)A 59 537 :M -.108(and \(viii\). )A f1_10 sf <5C>S 59 552 :M f2_10 sf 2.491 .249(Lemma )J 99 552 :M .835(9:)A f0_10 sf .502 .05( )J 114 552 :M -.078(If )A 125 552 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 137 552 :M .694 .069(is )J 149 552 :M .056 .006(a )J 158 552 :M .688 .069(minimal )J 197 552 :M -.142(d-connecting )A 252 552 :M .202 .02(path )J 274 552 :M -.116(between )A 311 552 :M .255 .026(X )J 323 552 :M -.313(and )A 341 552 :M .255 .026(Y )J 353 552 :M .189 .019(given )J 380 552 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 393 552 :M .601 .06(in )J 406 552 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 484 552 :M f4_10 sf (U)S 59 564 :M f0_10 sf .098 .01(contains )J 96 564 :M .218 .022(the )J 112 564 :M .232 .023(subpath )J 147 564 :M .255 .026(A )J 158 564 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 177 564 :M -.17(B )A 187 564 :M f1_10 sf .359A f0_10 sf .248 .025(* )J 206 564 :M .255 .026(D )J 217 564 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 236 564 :M 1.108 .111(C, )J 251 564 :M -.313(and )A 269 564 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 344 564 :M .098 .01(contains )J 382 564 :M .218 .022(the )J 399 564 :M -.344(edge )A 421 564 :M -.17(B )A 432 564 :M f1_10 sf .504A f0_10 sf .128 .013( )J 447 564 :M 1.108 .111(C, )J 462 564 :M .202 .02(then )J 484 564 :M f4_10 sf (U)S 59 576 :M f0_10 sf .138 .014(contains a unique subpath )J f4_10 sf .062(U)A f0_10 sf .125 .012(\(F,C\) that is a discriminating path for D.)J 59 591 :M f2_10 sf .216(Proof.)A f0_10 sf .492 .049( We will show that for each n )J cF f1_10 sf .049A sf .492 .049( 1, if )J 242 591 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 253 591 :M .098 .01(contains )J 290 591 :M .056 .006(a )J 298 591 :M -.081(vertex )A 326 591 :M .555 .056(M )J 339 591 :M .123 .012(such )J 361 591 :M .361 .036(that )J 380 591 :M f4_10 sf (U)S f0_10 sf (\(M,D\) )S 416 591 :M .694 .069(is )J 427 591 :M .144 .014(of )J 439 591 :M .315 .032(length )J 468 591 :M .833 .083(n, )J 480 591 :M -.328(for)A 59 603 :M -.065(every vertex Q )A 121 603 :M .417 .042(on )J 135 603 :M f4_10 sf (U)S f0_10 sf (\(M,D\) )S 171 603 :M -.099(except )A 200 603 :M -.052(for )A 215 603 :M .218 .022(the )J 231 603 :M -.043(endpoints )A 273 603 :M .255 .026(Q )J 284 603 :M .694 .069(is )J 295 603 :M .056 .006(a )J 303 603 :M -.13(collider )A 336 603 :M .417 .042(on )J 350 603 :M f4_10 sf .1(U)A f0_10 sf .261 .026(\(M,D\), )J 389 603 :M -.313(and )A 406 603 :M -.052(for )A 421 603 :M -.204(each )A 442 603 :M -.081(vertex )A 470 603 :M .255 .026(Q )J 481 603 :M (on)S 59 615 :M f4_10 sf -.065(U)A f0_10 sf -.043(\(M,D\) except )A 123 615 :M .647 .065(possibly )J 161 615 :M -.052(for )A 176 615 :M .255 .026(D )J 187 615 :M -.097(there )A 210 615 :M .694 .069(is )J 221 615 :M .051 .005(an )J 234 615 :M -.344(edge )A 255 615 :M .255 .026(Q )J 266 615 :M f1_10 sf .504A f0_10 sf .128 .013( )J 280 615 :M .755 .075(C )J 291 615 :M .601 .06(in )J 303 615 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 380 615 :M .202 .02(then )J 401 615 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 412 615 :M .098 .01(contains )J 449 615 :M .056 .006(a )J 457 615 :M -.081(vertex )A 485 615 :M (F)S 59 627 :M .196 .02(such that )J f4_10 sf .111(U)A f0_10 sf .169 .017(\(F,D\) is of length n + 1, and either )J f4_10 sf .111(U)A f0_10 sf .2 .02(\(F,D\) is a discriminating path for D, or )J f4_10 sf .111(U)A f0_10 sf .184 .018( contains an )J 474 627 :M -.626(edge)A 59 639 :M .124 .012(F )J f1_10 sf .192A f0_10 sf .263 .026( M, and MAG\()J f4_10 sf .133(G)A f0_10 sf .061<28>A f2_10 sf .143(O)A f0_10 sf (,)S f2_10 sf .103(S)A f0_10 sf (,)S f2_10 sf .123(L)A f0_10 sf .221 .022(\)\) contains an edge F )J f1_10 sf .182A f0_10 sf .179 .018( C.)J 59 654 :M .039 .004(By hypothesis, there is a path )J f4_10 sf (U)S f0_10 sf .037 .004(\(B,D\) of length 1 such that every vertex Q between B and D is )J 441 654 :M .056 .006(a )J 449 654 :M -.13(collider )A 482 654 :M (on)S 59 666 :M f4_10 sf .072(U)A f0_10 sf .099 .01( and there is an edge B )J f1_10 sf .098A f0_10 sf .121 .012( C in MAG\()J f4_10 sf .072(G)A f0_10 sf <28>S f2_10 sf .077(O)A f0_10 sf (,)S f2_10 sf .055(S)A f0_10 sf (,)S f2_10 sf .066(L)A f0_10 sf .045(\)\).)A endp %%Page: 6 6 %%BeginPageSetup initializepage (peter; page: 6 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (6)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf .141 .014(Suppose that )J f4_10 sf .062(U)A f0_10 sf .108 .011( contains a vertex )J 195 51 :M .555 .056(M )J 208 51 :M .123 .012(such )J 230 51 :M .361 .036(that )J 249 51 :M f4_10 sf (U)S f0_10 sf (\(M,D\) )S 285 51 :M .694 .069(is )J 296 51 :M .144 .014(of )J 308 51 :M .315 .032(length )J 337 51 :M .833 .083(n, )J 349 51 :M -.313(and )A 366 51 :M -.052(for )A 381 51 :M -.141(every )A 406 51 :M -.081(vertex )A 434 51 :M .255 .026(Q )J 445 51 :M .417 .042(on )J 459 51 :M f4_10 sf -.107(U)A f0_10 sf -.094(\(M,D\))A 59 63 :M -.03(except for the endpoints Q is a collider on )A f4_10 sf -.057(U)A f0_10 sf -.034(\(M,D\) and for each vertex Q )A 352 63 :M .417 .042(on )J 366 63 :M f4_10 sf (U)S f0_10 sf (\(M,D\) )S 402 63 :M -.099(except )A 431 63 :M .647 .065(possibly )J 469 63 :M -.052(for )A 484 63 :M (D)S 59 75 :M .112 .011(there is an edge Q )J f1_10 sf .102A f0_10 sf .125 .013( C in MAG\()J f4_10 sf .074(G)A f0_10 sf <28>S f2_10 sf .08(O)A f0_10 sf (,)S f2_10 sf .057(S)A f0_10 sf (,)S f2_10 sf .069(L)A f0_10 sf .153 .015(\)\). Because )J f4_10 sf .074(U)A f0_10 sf .143 .014( is a minimal d-connecting path, by Lemma 7 )J 472 75 :M -.246(there)A 59 87 :M .252 .025(is an edge F *)J f1_10 sf .242A f0_10 sf .18 .018( M on )J f4_10 sf .177(U)A f0_10 sf .281 .028(. The edge between F and M is either F )J f1_10 sf .242A f0_10 sf .195 .02( M, F o)J f1_10 sf .242A f0_10 sf .18 .018( M, or F )J 411 87 :M f1_10 sf .065A f0_10 sf ( )S 425 87 :M .926 .093(M. )J 441 87 :M -.078(If )A 451 87 :M f4_10 sf .208(U)A f0_10 sf .459 .046(\(F,C\) )J 484 87 :M .332(is)A 59 99 :M .015 .001(not a discriminating path for D then there is an edge )J 271 99 :M -.116(between )A 307 99 :M .855 .086(F )J 317 99 :M -.313(and )A 334 99 :M .755 .075(C )J 345 99 :M .601 .06(in )J 357 99 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\). )J 434 99 :M -.085(By )A 449 99 :M (Lemma )S 483 99 :M .5(7,)A 59 111 :M .048 .005(there are two cases: \(i\) the edge between F and C is out of C \(and hence C )J f1_10 sf S f0_10 sf .033 .003( F\), or \(ii\) )J 412 111 :M .218 .022(the )J 428 111 :M -.344(edge )A 449 111 :M -.116(between )A 485 111 :M (F)S 59 123 :M .14 .014(and C is F )J f1_10 sf .163A f0_10 sf .174 .017( C, and the edge between F and M is F )J f1_10 sf .172A f0_10 sf .191 .019( M.)J 59 138 :M .218 .022(If \(i\), there is no edge F o)J f1_10 sf .206A f0_10 sf .226 .023( M because C )J f1_10 sf .206A f0_10 sf .118 .012( F is )J 263 138 :M .674 .067(into )J 283 138 :M 1.201 .12(F. )J 296 138 :M -.078(If )A 306 138 :M .051 .005(\(i\) )J 319 138 :M -.313(and )A 336 138 :M .218 .022(the )J 352 138 :M -.344(edge )A 373 138 :M -.116(between )A 409 138 :M .855 .086(F )J 419 138 :M -.313(and )A 436 138 :M .555 .056(M )J 449 138 :M .979 .098(isF )J 466 138 :M f1_10 sf .504A f0_10 sf .128 .013( )J 480 138 :M .611(M,)A 59 150 :M -.02(then there )A 102 150 :M .694 .069(is )J 113 150 :M .056 .006(a )J 121 150 :M -.119(cycle )A 145 150 :M .855 .086(F )J 155 150 :M f1_10 sf .504A f0_10 sf .128 .013( )J 169 150 :M .555 .056(M )J 182 150 :M f1_10 sf .504A f0_10 sf .128 .013( )J 196 150 :M .755 .075(C )J 207 150 :M -.313(and )A 224 150 :M .755 .075(C )J 235 150 :M f1_10 sf .504A f0_10 sf .128 .013( )J 249 150 :M 1.201 .12(F. )J 262 150 :M -.078(If )A 272 150 :M .218 .022(the )J 288 150 :M -.344(edge )A 309 150 :M -.116(between )A 345 150 :M .855 .086(F )J 355 150 :M -.313(and )A 372 150 :M .555 .056(M )J 385 150 :M .694 .069(is )J 396 150 :M .855 .086(F )J 406 150 :M f1_10 sf .065A f0_10 sf ( )S 420 150 :M .555 .056(M )J 433 150 :M .202 .02(then )J 454 150 :M -.097(there )A 477 150 :M .694 .069(is )J 488 150 :M (a)S 59 162 :M .009 .001(contradiction because F is not an ancestor of M )J 252 162 :M .555 .056(but )J 269 162 :M -.097(there )A 292 162 :M .694 .069(is )J 303 162 :M .056 .006(a )J 311 162 :M .202 .02(path )J 332 162 :M .555 .056(M )J 345 162 :M f1_10 sf .504A f0_10 sf .128 .013( )J 359 162 :M .755 .075(C )J 370 162 :M f1_10 sf .504A f0_10 sf .128 .013( )J 384 162 :M 1.201 .12(F. )J 397 162 :M .328 .033(It )J 407 162 :M .298 .03(follows )J 441 162 :M .361 .036(that )J 460 162 :M -.176(case )A 480 162 :M -.071(\(ii\))A 59 174 :M .048 .005(holds. If )J f4_10 sf (U)S f0_10 sf .06 .006( does not contains a discriminating subpath, then for every subpath of )J f4_10 sf (U)S f0_10 sf .054 .005( there is a longer subpath)J 59 186 :M .119 .012(of )J f4_10 sf .104(U)A f0_10 sf .195 .02(. Because )J f4_10 sf .104(U)A f0_10 sf .195 .02( is of finite length, it follows that it contains a discriminating subpath )J f4_10 sf .104(U)A f0_10 sf .237 .024(\(M,C\) for D.)J 59 201 :M -.188(We )A 76 201 :M .676 .068(will )J 96 201 :M .216 .022(now )J 117 201 :M .28 .028(show )J 142 201 :M .361 .036(that )J 161 201 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 172 201 :M .694 .069(is )J 183 201 :M -.031(unique. )A 216 201 :M -.259(Because )A 251 201 :M .218 .022(the )J 267 201 :M -.344(edge )A 288 201 :M -.116(between )A 324 201 :M -.17(B )A 334 201 :M -.313(and )A 351 201 :M .755 .075(C )J 362 201 :M .694 .069(is )J 374 201 :M -.158(oriented )A 410 201 :M .144 .014(as )J 423 201 :M -.17(B )A 434 201 :M f1_10 sf .504A f0_10 sf .128 .013( )J 449 201 :M 1.108 .111(C, )J 464 201 :M .555 .056(M )J 478 201 :M .039(lies)A 59 213 :M -.116(between )A 95 213 :M .255 .026(X )J 106 213 :M -.313(and )A 123 213 :M 1.108 .111(C. )J 137 213 :M .234 .023(No )J 153 213 :M .232 .023(subpath )J 188 213 :M .144 .014(of )J 200 213 :M f4_10 sf .119(U)A f0_10 sf .298 .03(\(M,C\) )J 236 213 :M .694 .069(is )J 247 213 :M .056 .006(a )J 255 213 :M -.019(discriminating )A 317 213 :M .202 .02(path )J 339 213 :M -.052(for )A 355 213 :M .255 .026(D )J 367 213 :M -.163(because )A 402 213 :M .388 .039(all )J 417 213 :M .144 .014(of )J 430 213 :M .218 .022(the )J 447 213 :M -.074(vertices )A 482 213 :M (on)S 59 225 :M f4_10 sf .056(U)A f0_10 sf .109 .011(\(M,C\) except for M are adjacent to C. No path containing )J f4_10 sf .056(U)A f0_10 sf .101 .01(\(M,C\) is )J 346 225 :M .056 .006(a )J 354 225 :M -.019(discriminating )A 415 225 :M .202 .02(path )J 436 225 :M -.052(for )A 451 225 :M .255 .026(D )J 462 225 :M -.274(because)A 59 237 :M .085 .008(M is not adjacent to C. )J f1_10 sf <5C>S 59 252 :M f0_10 sf .144 .014(In )J 71 252 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 148 252 :M .056 .006(a )J 156 252 :M -.081(vertex )A 184 252 :M .255 .026(V )J 195 252 :M .694 .069(is )J 206 252 :M .056 .006(a )J 214 252 :M f2_10 sf 1.904 .19(hidden )J 250 252 :M .588(vertex)A f0_10 sf .331 .033( )J 284 252 :M .417 .042(on )J 298 252 :M .056 .006(a )J 306 252 :M -.019(discriminating )A 367 252 :M .202 .02(path )J 388 252 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 399 252 :M .328 .033(if )J 409 252 :M -.313(and )A 426 252 :M .515 .052(only )J 448 252 :M .328 .033(if )J 458 252 :M -.097(there )A 481 252 :M -.602(are)A 59 264 :M -.074(vertices )A 94 264 :M .255 .026(X )J 106 264 :M -.313(and )A 125 264 :M .255 .026(Y )J 138 264 :M .417 .042(on )J 154 264 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 167 264 :M .123 .012(such )J 191 264 :M .361 .036(that )J 212 264 :M f4_10 sf .987(V)A f0_10 sf .404 .04( )J 225 264 :M .694 .069(is )J 238 264 :M -.226(adjacent )A 275 264 :M .601 .06(to )J 289 264 :M .255 .026(X )J 302 264 :M -.313(and )A 321 264 :M .255 .026(Y )J 334 264 :M .417 .042(on )J 350 264 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 366 264 :M -.313(and )A 385 264 :M .255 .026(X )J 398 264 :M -.313(and )A 417 264 :M .255 .026(Y )J 430 264 :M -.235(are )A 447 264 :M -.226(adjacent )A 484 264 :M .222(in)A 59 276 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .105(\)\).)A 59 291 :M f2_10 sf 2.223 .222(Lemma 10: )J f0_10 sf 1.178 .118(In )J 130 291 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 207 291 :M .328 .033(if )J 217 291 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 228 291 :M .694 .069(is )J 239 291 :M .056 .006(a )J 247 291 :M .688 .069(minimal )J 285 291 :M -.142(d-connecting )A 339 291 :M .202 .02(path )J 360 291 :M -.116(between )A 396 291 :M .255 .026(X )J 407 291 :M -.313(and )A 424 291 :M .255 .026(Y )J 435 291 :M .189 .019(given )J 461 291 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(, )J 475 291 :M -.072(then)A 59 303 :M .05 .005(there is no pair of distinct vertices B, J such that B is a hidden )J 311 303 :M -.081(vertex )A 339 303 :M .417 .042(on )J 353 303 :M .218 .022(the )J 369 303 :M -.019(discriminating )A 430 303 :M .202 .02(path )J 451 303 :M f4_10 sf -.103(U)A f0_10 sf -.064(\(I,K\) )A 481 303 :M -.328(for)A 59 315 :M .009 .001(J, and J is a hidden vertex on the discriminating path )J f4_10 sf (U)S f0_10 sf (\(A,C\) for B.)S 59 330 :M f2_10 sf .172(Proof.)A f0_10 sf .497 .05( Suppose on the contrary that B )J 221 330 :M .694 .069(is )J 232 330 :M .056 .006(a )J 240 330 :M -.286(hidden )A 269 330 :M -.081(vertex )A 297 330 :M .417 .042(on )J 311 330 :M f4_10 sf (U)S f0_10 sf .038 .004(\(I,K\), )J 344 330 :M -.313(and )A 361 330 :M .555 .056(J )J 369 330 :M .694 .069(is )J 380 330 :M .056 .006(a )J 388 330 :M -.286(hidden )A 417 330 :M -.081(vertex )A 445 330 :M .417 .042(on )J 459 330 :M f4_10 sf .052(U)A f0_10 sf .037(\(A,C\).)A 59 342 :M .146 .015(Because B is hidden on )J f4_10 sf .08(U)A f0_10 sf .122 .012(\(I,K\), C lies on )J f4_10 sf .08(U)A f0_10 sf .13 .013(\(I,K\). C )J f1_10 sf .061A f0_10 sf .14 .014(\312K because B lies )J 348 342 :M .417 .042(on )J 362 342 :M f4_10 sf (U)S f0_10 sf .038 .004(\(I,K\), )J 395 342 :M .218 .022(the )J 411 342 :M .515 .052(only )J 433 342 :M -.081(vertex )A 461 342 :M -.33(adjacent)A 59 354 :M .601 .06(to )J 71 354 :M .255 .026(K )J 82 354 :M .417 .042(on )J 96 354 :M f4_10 sf -.103(U)A f0_10 sf -.064(\(I,K\) )A 126 354 :M .5 .05( )J 130 354 :M .694 .069(is )J 141 354 :M .926 .093(J, )J 152 354 :M -.313(and )A 169 354 :M -.17(B )A 179 354 :M f1_10 sf 1.341A f0_10 sf 2.14 .214(\312J. )J 201 354 :M .755 .075(C )J 212 354 :M f1_10 sf 1.383A f0_10 sf 1.866 .187J 231 354 :M -.163(because )A 265 354 :M -.041(otherwise )A 307 354 :M .555 .056(J )J 315 354 :M .694 .069(is )J 327 354 :M .555 .056(not )J 345 354 :M .056 .006(a )J 354 354 :M -.286(hidden )A 384 354 :M -.081(vertex )A 413 354 :M .417 .042(on )J 428 354 :M f4_10 sf .151(U)A f0_10 sf .366 .037(\(A,C\). )J 466 354 :M .755 .075(C )J 478 354 :M f1_10 sf 1.302A f0_10 sf 1.382A 59 366 :M -.163(because )A 93 366 :M -.041(otherwise )A 135 366 :M .926 .093(J, )J 146 366 :M .043 .004(which )J 174 366 :M .694 .069(is )J 185 366 :M .2(on)A f4_10 sf .1 .01( )J 199 366 :M .052(U)A f0_10 sf .123 .012(\(A,C\) )J 233 366 :M .555 .056(but )J 250 366 :M .555 .056(not )J 267 366 :M -.231(equal )A 291 366 :M .601 .06(to )J 303 366 :M -.17(B )A 313 366 :M .144 .014(or )J 325 366 :M .255 .026(A )J 336 366 :M .694 .069(is )J 347 366 :M .051 .005(an )J 360 366 :M -.101(ancestor )A 396 366 :M .144 .014(of )J 408 366 :M .755 .075(C )J 419 366 :M -.14(= )A 428 366 :M .56 .056(I, )J 438 366 :M -.313(and )A 455 366 :M -.163(hence )A 482 366 :M (by)S 59 378 :M -.295(repeated )A 95 378 :M .083 .008(applications )J 148 378 :M .144 .014(of )J 161 378 :M (Lemma )S 196 378 :M .454 .045(1 )J 206 378 :M .232 .023(through )J 242 378 :M (Lemma )S 278 378 :M .454 .045(5 )J 289 378 :M -.097(there )A 314 378 :M .694 .069(is )J 327 378 :M .051 .005(an )J 342 378 :M -.062(inducing )A 382 378 :M .202 .02(path )J 405 378 :M -.116(between )A 443 378 :M .157 .016(I )J 452 378 :M -.313(and )A 471 378 :M .255 .026(K )J 484 378 :M .222(in)A 59 390 :M f4_10 sf .091(G)A f0_10 sf <28>S f2_10 sf .098(O)A f0_10 sf (,)S f2_10 sf .07(S)A f0_10 sf (,)S f2_10 sf .084(L)A f0_10 sf .169 .017(\). But by definition of discriminating path I and K )J 301 390 :M -.235(are )A 316 390 :M .555 .056(not )J 333 390 :M -.226(adjacent )A 368 390 :M .601 .06(in )J 380 390 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\). )J 457 390 :M -.207(Hence )A 485 390 :M (C)S 59 402 :M f1_10 sf .177A f0_10 sf .359 .036(\312I. Because K is on )J f4_10 sf .233(U)A f0_10 sf .356 .036(\(A,C\) but not equal to A, B, or C, )J 295 402 :M -.097(there )A 318 402 :M .694 .069(is )J 329 402 :M .056 .006(a )J 337 402 :M -.337(directed )A 370 402 :M .202 .02(path )J 391 402 :M .047 .005(from )J 414 402 :M .255 .026(K )J 425 402 :M .601 .06(to )J 437 402 :M 1.108 .111(C. )J 451 402 :M .143(Similarly,)A 59 414 :M -.097(there )A 82 414 :M .694 .069(is )J 93 414 :M .056 .006(a )J 101 414 :M -.337(directed )A 134 414 :M .202 .02(path )J 156 414 :M .047 .005(from )J 180 414 :M .755 .075(C )J 192 414 :M .601 .06(to )J 205 414 :M .65 .065(K. )J 220 414 :M -.089(Hence, )A 252 414 :M -.097(there )A 276 414 :M .694 .069(is )J 288 414 :M .056 .006(a )J 297 414 :M -.337(directed )A 331 414 :M -.119(cycle )A 356 414 :M .601 .06(in )J 369 414 :M .193(MAG\()A f4_10 sf .209(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .161(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .259 .026(\)\), )J 447 414 :M .043 .004(which )J 476 414 :M .694 .069(is )J 488 414 :M (a)S 59 426 :M -.138(contradiction. )A f1_10 sf <5C>S 59 441 :M f2_10 sf 1.443 .144(Lemma 11:)J f0_10 sf .662 .066( In a MAG\()J f4_10 sf .407(G)A f0_10 sf .188<28>A f2_10 sf .439(O)A f0_10 sf .141(,)A f2_10 sf .314(S)A f0_10 sf .141(,)A f2_10 sf .376(L)A f0_10 sf .423 .042(\)\), if )J f4_10 sf .407(U)A f0_10 sf .58 .058( is a minimal )J 285 441 :M -.142(d-connecting )A 339 441 :M .202 .02(path )J 360 441 :M -.116(between )A 396 441 :M .255 .026(X )J 407 441 :M -.313(and )A 424 441 :M .255 .026(Y )J 435 441 :M .189 .019(given )J 461 441 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(, )J 475 441 :M -.072(then)A 59 453 :M -.097(there )A 83 453 :M .694 .069(is )J 95 453 :M .417 .042(no )J 110 453 :M .249 .025(triple )J 136 453 :M .144 .014(of )J 149 453 :M .033 .003(distinct )J 183 453 :M -.286(hidden )A 213 453 :M -.074(vertices )A 249 453 :M .65 .065(X, )J 265 453 :M .65 .065(Y, )J 281 453 :M .356 .036(Z )J 293 453 :M .417 .042(on )J 309 453 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 322 453 :M .123 .012(such )J 346 453 :M .361 .036(that )J 367 453 :M .255 .026(X )J 380 453 :M .694 .069(is )J 393 453 :M .056 .006(a )J 403 453 :M -.286(hidden )A 434 453 :M -.081(vertex )A 464 453 :M .417 .042(on )J 480 453 :M -.108(the)A 59 465 :M -.019(discriminating )A 120 465 :M .202 .02(path )J 141 465 :M -.052(for )A 156 465 :M .255 .026(Y )J 167 465 :M .417 .042(on )J 181 465 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 195 465 :M .255 .026(Y )J 206 465 :M .694 .069(is )J 217 465 :M .056 .006(a )J 225 465 :M -.286(hidden )A 254 465 :M -.081(vertex )A 283 465 :M .417 .042(on )J 298 465 :M .218 .022(the )J 315 465 :M -.019(discriminating )A 377 465 :M .202 .02(path )J 399 465 :M -.052(for )A 415 465 :M .356 .036(Z )J 426 465 :M .417 .042(on )J 441 465 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 456 465 :M -.313(and )A 474 465 :M .356 .036(Z )J 485 465 :M .332(is)A 59 477 :M -.019(between Y and X on )A f4_10 sf (U)S f0_10 sf (.)S 59 492 :M f2_10 sf .056(Proof.)A f0_10 sf .094 .009( Let )J f4_10 sf .092(U)A f4_6 sf 0 2 rm (Y)S 0 -2 rm f0_10 sf .159 .016( be the discriminating path for Y on )J f4_10 sf .092(U)A f0_10 sf .158 .016(, and similarly for )J f4_10 sf .092(U)A f4_6 sf 0 2 rm (Z)S 0 -2 rm f0_10 sf .148 .015(. Because X is a hidden )J 454 492 :M -.081(vertex )A 482 492 :M (on)S 59 504 :M .053 .005(the discriminating path for Y on )J f4_10 sf (U)S f0_10 sf .049 .005(, every vertex between X and Y is on the discriminating path for Y on )J 482 504 :M f4_10 sf .209(U)A f0_10 sf (.)S 59 516 :M .045 .005(Hence Z is on the discriminating path for Y on )J f4_10 sf (U)S f0_10 sf .041 .004(. Because Z is between Y and X, and neither Y nor X is )J 483 516 :M -.439(an)A 59 528 :M .242 .024(endpoint of )J f4_10 sf .116(U)A f4_6 sf 0 2 rm .053(Y)A 0 -2 rm f0_10 sf .164 .016(, Z is not an endpoint of )J f4_10 sf .116(U)A f4_6 sf 0 2 rm .053(Y)A 0 -2 rm f0_10 sf .175 .017(; it follows )J 277 528 :M .361 .036(that )J 296 528 :M -.204(each )A 317 528 :M .144 .014(of )J 329 528 :M .218 .022(the )J 345 528 :M -.074(vertices )A 379 528 :M -.226(adjacent )A 414 528 :M .601 .06(to )J 426 528 :M .356 .036(Z )J 436 528 :M .417 .042(on )J 450 528 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 461 528 :M -.235(are )A 476 528 :M -.035(also)A 59 540 :M .417 .042(on )J 74 540 :M f4_10 sf .207(U)A f4_6 sf 0 2 rm .095(Y)A 0 -2 rm f0_10 sf .13 .013(. )J 92 540 :M -.259(Because )A 128 540 :M .356 .036(Z )J 139 540 :M .694 .069(is )J 151 540 :M .056 .006(a )J 160 540 :M -.286(hidden )A 190 540 :M -.081(vertex )A 219 540 :M .417 .042(on )J 235 540 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 251 540 :M -.313(and )A 270 540 :M .515 .052(both )J 294 540 :M .144 .014(of )J 308 540 :M .218 .022(the )J 326 540 :M -.074(vertices )A 362 540 :M -.226(adjacent )A 399 540 :M .601 .06(to )J 413 540 :M .356 .036(Z )J 425 540 :M -.235(are )A 442 540 :M .281 .028(also )J 464 540 :M .417 .042(on )J 480 540 :M -.108(the)A 59 552 :M .166 .017(discriminating path for Y on )J f4_10 sf .082(U)A f0_10 sf .117 .012(, Z is a hidden vertex on )J f4_10 sf .082(U)A f4_6 sf 0 2 rm (Y)S 0 -2 rm f0_10 sf .126 .013(. By hypothesis, Y is a )J 390 552 :M -.286(hidden )A 419 552 :M -.081(vertex )A 447 552 :M .417 .042(on )J 461 552 :M f4_10 sf .207(U)A f4_6 sf 0 2 rm .095(Z)A 0 -2 rm f0_10 sf .13 .013(. )J 478 552 :M -.224(But)A 59 564 :M .098 .01(this contradicts Lemma 10. )J 171 552 9 12 rC gS 1.286 1.111 scale 133.001 506.705 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 59 579 :M f2_10 sf 1.443 .144(Lemma 12:)J f0_10 sf .662 .066( In a MAG\()J f4_10 sf .407(G)A f0_10 sf .188<28>A f2_10 sf .439(O)A f0_10 sf .141(,)A f2_10 sf .314(S)A f0_10 sf .141(,)A f2_10 sf .376(L)A f0_10 sf .423 .042(\)\), if )J f4_10 sf .407(U)A f0_10 sf .58 .058( is a minimal )J 285 579 :M -.142(d-connecting )A 339 579 :M .202 .02(path )J 360 579 :M -.116(between )A 396 579 :M .255 .026(X )J 407 579 :M -.313(and )A 424 579 :M .255 .026(Y )J 435 579 :M .189 .019(given )J 461 579 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(, )J 475 579 :M -.072(then)A 59 591 :M -.019(there is no quadruple of distinct hidden vertices A)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf -.02(, A)A f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf -.02(, A)A f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf -.02(, A)A f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf -.019( in that order )A 369 591 :M .417 .042(on )J 383 591 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 394 591 :M .123 .012(such )J 416 591 :M .361 .036(that )J 435 591 :M .389(A)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 448 591 :M .694 .069(is )J 459 591 :M .056 .006(a )J 467 591 :M -.443(hidden)A 59 603 :M .01 .001(vertex on the discriminating path for A)J f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf ( on )S f4_10 sf (U)S f0_10 sf (, and A)S f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf .008 .001( is a hidden vertex on the discriminating )J 441 603 :M .202 .02(path )J 462 603 :M -.052(for )A 477 603 :M -.277(A)A f0_6 sf 0 2 rm -.161(r+1)A 0 -2 rm 59 615 :M f0_10 sf .338 .034(on )J f4_10 sf .254(U)A f0_10 sf (.)S 59 630 :M f2_10 sf .33(Proof.)A f0_10 sf .551 .055( Let )J f4_10 sf .543(U)A f0_6 sf 0 2 rm .202(i+1)A 0 -2 rm f0_10 sf .472 .047( be )J 140 630 :M .218 .022(the )J 156 630 :M -.019(discriminating )A 217 630 :M .202 .02(path )J 238 630 :M -.052(for )A 253 630 :M .093(A)A f0_6 sf 0 2 rm .035(i+1)A 0 -2 rm f0_10 sf ( )S 272 630 :M .417 .042(on )J 286 630 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 300 630 :M -.313(and )A 317 630 :M .502 .05(similarly )J 357 630 :M -.052(for )A 372 630 :M f4_10 sf .14(U)A f0_6 sf 0 2 rm .054(r+1)A 0 -2 rm f0_10 sf .088 .009(. )J 394 630 :M .361 .036(Suppose )J 432 630 :M -.101(contrary )A 468 630 :M .601 .06(to )J 480 630 :M -.108(the)A 59 642 :M -.003(hypothesis there is a quadruple of distinct hidden vertices A)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf (, A)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf (, A)S f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf (, A)S f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf -.003( in that order on )A f4_10 sf (U)S f0_10 sf ( such )S 453 642 :M .361 .036(that )J 472 642 :M .389(A)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 485 642 :M .332(is)A 59 654 :M -.015(a hidden vertex on the discriminating path for A)A f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf -.014( on )A f4_10 sf (U)S f0_10 sf -.016(, and A)A f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf -.015( is a hidden vertex on the discriminating path)A 59 666 :M .123 .012(for A)J f0_6 sf 0 2 rm .024(r+1)A 0 -2 rm f0_10 sf .054 .005( on )J f4_10 sf .062(U)A f0_10 sf .081 .008(. A)J f0_6 sf 0 2 rm .024(r+1)A 0 -2 rm f0_10 sf .056 .006( is on )J f4_10 sf .062(U)A f0_6 sf 0 2 rm .023(i+1)A 0 -2 rm f0_10 sf .113 .011( because it is between A)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .088 .009( and A)J f0_6 sf 0 2 rm .023(i+1)A 0 -2 rm f0_10 sf .093 .009(, and A)J f0_6 sf 0 2 rm (i )S 0 -2 rm 340 666 :M f0_10 sf .694 .069(is )J 351 666 :M .417 .042(on )J 365 666 :M f4_10 sf .26(U)A f0_6 sf 0 2 rm .097(i+1)A 0 -2 rm f0_10 sf .164 .016(. )J 387 666 :M (A)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf ( )S 406 666 :M .694 .069(is )J 417 666 :M .056 .006(a )J 425 666 :M -.286(hidden )A 454 666 :M -.081(vertex )A 482 666 :M (on)S 59 678 :M f4_10 sf (U)S f0_10 sf .047 .005(, and both of the vertices )J 169 678 :M -.226(adjacent )A 204 678 :M .601 .06(to )J 216 678 :M (A)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf ( )S 235 678 :M .417 .042(on )J 249 678 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 260 678 :M -.235(are )A 275 678 :M .281 .028(also )J 295 678 :M .417 .042(on )J 309 678 :M f4_10 sf .26(U)A f0_6 sf 0 2 rm .097(i+1)A 0 -2 rm f0_10 sf .164 .016(, )J 331 678 :M -.163(because )A 365 678 :M (A)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf ( )S 384 678 :M .694 .069(is )J 395 678 :M .555 .056(not )J 412 678 :M .051 .005(an )J 425 678 :M -.062(endpoint )A 463 678 :M .144 .014(of )J 475 678 :M f4_10 sf .099(U)A f0_6 sf 0 2 rm .044(i+1.)A 0 -2 rm endp %%Page: 7 7 %%BeginPageSetup initializepage (peter; page: 7 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (7)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf (Hence A)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf .013 .001( is a hidden vertex on )J f4_10 sf (U)S f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf .02 .002(. Similarly, A)J f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf .013 .001( is a hidden vertex on )J f4_10 sf (U)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf (. )S 379 51 :M .04 .004(But )J 397 51 :M .753 .075(this )J 416 51 :M -.124(contradicts )A 462 51 :M -.135(Lemma)A 59 63 :M .25(10.)A 59 78 :M f2_10 sf 1.443 .144(Lemma 13:)J f0_10 sf .662 .066( In a MAG\()J f4_10 sf .407(G)A f0_10 sf .188<28>A f2_10 sf .439(O)A f0_10 sf .141(,)A f2_10 sf .314(S)A f0_10 sf .141(,)A f2_10 sf .376(L)A f0_10 sf .423 .042(\)\), if )J f4_10 sf .407(U)A f0_10 sf .58 .058( is a minimal )J 285 78 :M -.142(d-connecting )A 339 78 :M .202 .02(path )J 360 78 :M -.116(between )A 396 78 :M .255 .026(X )J 407 78 :M -.313(and )A 424 78 :M .255 .026(Y )J 435 78 :M .189 .019(given )J 461 78 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(, )J 475 78 :M -.072(then)A 59 90 :M .125 .012(there is )J 92 90 :M .417 .042(no )J 106 90 :M -.268(sequence )A 144 90 :M .144 .014(of )J 156 90 :M .315 .032(length )J 185 90 :M -.178(greater )A 215 90 :M .202 .02(than )J 236 90 :M .454 .045(1 )J 245 90 :M .144 .014(of )J 257 90 :M .033 .003(distinct )J 290 90 :M -.074(vertices )A 324 90 :M .385( )J 388 90 :M .123 .012(such )J 410 90 :M .361 .036(that )J 429 90 :M -.052(for )A 444 90 :M -.204(each )A 465 90 :M (pair )S 484 90 :M -.328(of)A 59 102 :M -.023(vertices A)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf -.023(, A)A f0_6 sf 0 2 rm (i+1)S 0 -2 rm f0_10 sf -.022( that are adjacent in the sequence, A)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf -.022( is a hidden vertex on the discriminating )A 430 102 :M .202 .02(path )J 451 102 :M .144 .014(of )J 463 102 :M .093(A)A f0_6 sf 0 2 rm .035(i+1)A 0 -2 rm f0_10 sf ( )S 482 102 :M (on)S 59 114 :M f4_10 sf (U)S f0_10 sf .062 .006(, and A)J f0_6 sf 0 2 rm (n)S 0 -2 rm f0_10 sf .076 .008( is a hidden vertex on the discriminating path of A)J f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .036 .004( on )J f4_10 sf (U)S f0_10 sf .064 .006(. \(Note that the )J 390 114 :M .255 .025(subscripts )J 434 114 :M .144 .014(of )J 446 114 :M .218 .022(the )J 462 114 :M -.156(vertices)A 59 126 :M -.035(do not necessarily reflect the order in which they occur on )A f4_10 sf -.064(U)A f0_10 sf -.052(.\))A 59 141 :M f2_10 sf .135(Proof.)A f0_10 sf .408 .041( Suppose without loss of generality that n is greater than 1, and A)J f0_6 sf 0 2 rm .092(1)A 0 -2 rm f0_10 sf .289 .029( is to the right of A)J f0_6 sf 0 2 rm .092(n)A 0 -2 rm f0_10 sf .077 .008( )J 444 141 :M .417 .042(on )J 458 141 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 472 141 :M .134 .013(Let )J 489 141 :M (r)S 59 153 :M .268 .027(be the highest index such that A)J f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf .168 .017( is to the right of )J 263 153 :M .159(A)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 277 153 :M .328 .033(if )J 287 153 :M .123 .012(such )J 309 153 :M .056 .006(a )J 317 153 :M -.081(vertex )A 345 153 :M .558 .056(exists; )J 375 153 :M -.041(otherwise )A 417 153 :M .388 .039(let )J 431 153 :M .157 .016(r )J 438 153 :M -.14(= )A 447 153 :M .833 .083(1. )J 459 153 :M -.188(We )A 476 153 :M .149(will)A 59 165 :M .184 .018(now show that A)J f0_6 sf 0 2 rm .034(r+1)A 0 -2 rm f0_10 sf .107 .011( is to the left of A)J f0_6 sf 0 2 rm (n)S 0 -2 rm f0_10 sf .136 .014(. If r = 1, every vertex except A)J f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf ( )S 345 165 :M .694 .069(is )J 356 165 :M .601 .06(to )J 368 165 :M .218 .022(the )J 384 165 :M .126 .013(left )J 401 165 :M .144 .014(of )J 413 165 :M .37(A)A f0_6 sf 0 2 rm .154(1)A 0 -2 rm f0_10 sf .233 .023(, )J 430 165 :M .509 .051(so )J 443 165 :M .159(A)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 457 165 :M .694 .069(is )J 468 165 :M .601 .06(to )J 480 165 :M -.108(the)A 59 177 :M .194 .019(left of A)J f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .179 .018(, and A)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf ( )S f1_10 sf .091A f0_10 sf .135 .013( A)J f0_6 sf 0 2 rm (n)S 0 -2 rm f0_10 sf .206 .021( by Lemma 10. By Lemma 11 then A)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .204 .02( is not between A)J f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .17 .017( and A)J f0_6 sf 0 2 rm (n)S 0 -2 rm f0_10 sf .113 .011(, so )J 430 177 :M .786 .079(it )J 440 177 :M .694 .069(is )J 451 177 :M .601 .06(to )J 463 177 :M .218 .022(the )J 479 177 :M -.108(left)A 59 189 :M .265 .026(of A)J f0_6 sf 0 2 rm .062(n)A 0 -2 rm f0_10 sf .111 .011(. If r )J f1_10 sf .113A f0_10 sf .203 .02( 1, then A)J f0_6 sf 0 2 rm .057(r+1)A 0 -2 rm f0_10 sf .047 .005( )J f1_10 sf .113A f0_10 sf .166 .017( A)J f0_6 sf 0 2 rm .062(n)A 0 -2 rm f0_10 sf .26 .026( by Lemma 11, and A)J f0_6 sf 0 2 rm .057(r+1)A 0 -2 rm f0_10 sf .235 .023( is not between )J 339 189 :M .159(A)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 353 189 :M -.313(and )A 370 189 :M .159(A)A f0_6 sf 0 2 rm .066(n)A 0 -2 rm f0_10 sf .055 .006( )J 384 189 :M .417 .042(by )J 398 189 :M (Lemma )S 432 189 :M .769 .077(12. )J 449 189 :M -.207(Hence )A 477 189 :M -.277(A)A f0_6 sf 0 2 rm -.161(r+1)A 0 -2 rm 59 201 :M f0_10 sf .331 .033(is to the left of A)J f0_6 sf 0 2 rm .101(n)A 0 -2 rm f0_10 sf (.)S 59 216 :M .089 .009(We will now show that some vertex A)J f0_6 sf 0 2 rm .018(s+1)A 0 -2 rm f0_10 sf .071 .007( whose )J 254 216 :M -.143(index )A 279 216 :M .694 .069(is )J 290 216 :M -.178(greater )A 320 216 :M .202 .02(than )J 341 216 :M .41 .041(r+1 )J 359 216 :M .694 .069(is )J 370 216 :M .601 .06(to )J 382 216 :M .218 .022(the )J 398 216 :M .41 .041(right )J 421 216 :M .144 .014(of )J 433 216 :M .398(A)A f0_6 sf 0 2 rm .11(r)A 0 -2 rm f0_10 sf .251 .025(. )J 449 216 :M (A)S f0_6 sf 0 2 rm (r+2)S 0 -2 rm f0_10 sf ( )S 468 216 :M .694 .069(is )J 479 216 :M .111(not)A 59 228 :M -.027(between A)A f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf -.025( and A)A f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf -.024( by Lemma 11, and hence not equal to A)A f0_6 sf 0 2 rm (n)S 0 -2 rm f0_10 sf -.023(. There are two cases. )A 393 228 :M -.078(If )A 403 228 :M (A)S f0_6 sf 0 2 rm (r+2)S 0 -2 rm f0_10 sf ( )S 422 228 :M .694 .069(is )J 433 228 :M .601 .06(to )J 445 228 :M .218 .022(the )J 461 228 :M .41 .041(right )J 484 228 :M -.328(of)A 59 240 :M .101(A)A f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf .187 .019(, then we are done. Suppose then that A)J f0_6 sf 0 2 rm .039(r+2)A 0 -2 rm f0_10 sf .124 .012( is to the left of A)J f0_6 sf 0 2 rm .039(r+1)A 0 -2 rm f0_10 sf .064 .006(. )J 324 240 :M .328 .033(It )J 334 240 :M .298 .03(follows )J 368 240 :M .361 .036(that )J 387 240 :M -.097(there )A 410 240 :M .694 .069(is )J 421 240 :M .281 .028(some )J 446 240 :M -.081(vertex )A 474 240 :M .075(with)A 59 252 :M .163 .016(index greater than r+2 \(e.g. A)J f0_6 sf 0 2 rm (n)S 0 -2 rm f0_10 sf .121 .012(\) on the other side of A)J f0_6 sf 0 2 rm .031(r+1)A 0 -2 rm f0_10 sf .116 .012(. It follows that for some s > r such that )J 447 252 :M (A)S f0_6 sf 0 2 rm (s)S 0 -2 rm f0_10 sf ( )S 460 252 :M -.313(and )A 477 252 :M -.424(A)A f0_6 sf 0 2 rm -.256(s+1)A 0 -2 rm 59 264 :M f0_10 sf .053 .005(are on opposite sides of A)J f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf .054 .005( \(where A)J f0_6 sf 0 2 rm (s)S 0 -2 rm f0_10 sf .034 .003( is to the left of A)J f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf .043 .004(.\) A)J f0_6 sf 0 2 rm (s+1)S 0 -2 rm f0_10 sf .044 .004( is not between )J 382 264 :M (A)S f0_6 sf 0 2 rm (r+1)S 0 -2 rm f0_10 sf ( )S 401 264 :M -.313(and )A 418 264 :M .175(A)A f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf .061 .006( )J 431 264 :M .417 .042(by )J 445 264 :M (Lemma )S 479 264 :M .25(12,)A 59 276 :M .361 .036(and hence A)J f0_6 sf 0 2 rm .065(s+1)A 0 -2 rm f0_10 sf .051 .005( )J f1_10 sf .123A f0_10 sf .181 .018( A)J f0_6 sf 0 2 rm .067(n)A 0 -2 rm f0_10 sf .218 .022(. So A)J f0_6 sf 0 2 rm .065(s+1)A 0 -2 rm f0_10 sf .21 .021( is to the right of A)J f0_6 sf 0 2 rm (r)S 0 -2 rm f0_10 sf .189 .019(. But this is a )J 312 276 :M -.054(contradiction, )A 370 276 :M -.163(because )A 404 276 :M .157 .016(r )J 411 276 :M .694 .069(is )J 422 276 :M .218 .022(the )J 438 276 :M .362 .036(highest )J 471 276 :M -.304(index)A 59 288 :M .391 .039(such that A)J f0_6 sf 0 2 rm .053(r)A 0 -2 rm f0_10 sf .25 .025( is to the right of A)J f0_6 sf 0 2 rm (1)S 0 -2 rm 189 276 9 12 rC gS 1.286 1.111 scale 147.001 258.303 :M f1_10 sf <5C>S gR gR gS 0 0 552 730 rC 59 303 :M f0_10 sf -.188(We )A 76 303 :M .676 .068(will )J 96 303 :M .216 .022(now )J 117 303 :M -.083(recursively )A 164 303 :M -.247(define )A 191 303 :M .218 .022(the )J 207 303 :M -.319(order )A 230 303 :M .144 .014(of )J 242 303 :M .056 .006(a )J 250 303 :M -.019(discriminating )A 311 303 :M .202 .02(path )J 332 303 :M -.052(for )A 347 303 :M .056 .006(a )J 356 303 :M -.286(hidden )A 386 303 :M -.087(variable )A 422 303 :M .417 .042(on )J 437 303 :M .056 .006(a )J 446 303 :M .688 .069(minimal )J 485 303 :M -1.328(d-)A 59 315 :M .031 .003(connecting path. If )J f4_10 sf (U)S f0_10 sf .027 .003( is a minimal d-connecting path between X and Y given )J f2_10 sf (Z)S f0_10 sf (, and )S 400 315 :M .056 .006(W )J 413 315 :M .694 .069(is )J 424 315 :M .056 .006(a )J 432 315 :M -.286(hidden )A 461 315 :M -.171(variable)A 59 327 :M .048 .005(on )J f4_10 sf (U)S f0_10 sf .065 .007( such that the discriminating path for W on )J f4_10 sf (U)S f0_10 sf .068 .007( contains no hidden )J 343 327 :M -.065(variables )A 382 327 :M (other )S 406 327 :M .202 .02(than )J 427 327 :M .468 .047(W, )J 443 327 :M .202 .02(then )J 464 327 :M .056 .006(W )J 477 327 :M .694 .069(is )J 488 327 :M (a)S 59 339 :M f2_10 sf .733 .073(0-order hidden variable)J f0_10 sf .194 .019( on )J f4_10 sf .224(U)A f0_10 sf .175 .018(. If )J f4_10 sf .224(U)A f0_10 sf .427 .043( is a minimal d-connecting path between X and Y given )J f2_10 sf .207(Z)A f0_10 sf .272 .027(, and )J 472 339 :M .056 .006(W )J 485 339 :M .332(is)A 59 351 :M -.035(a hidden variable on )A f4_10 sf -.063(U)A f0_10 sf -.036( such that the maximum order of any other hidden variable on )A 398 351 :M .218 .022(the )J 414 351 :M -.019(discriminating )A 475 351 :M -.072(path)A 59 363 :M .565 .056(for W on )J f4_10 sf .412(U)A f0_10 sf .381 .038( is n)J f1_10 sf .313(-)A f0_10 sf .543 .054(1, then W is an )J f2_10 sf .317(n)A f2_6 sf 0 -4 rm .152(th)A 0 4 rm f2_10 sf 1.313 .131(-order hidden variable)J f0_10 sf .357 .036( on )J f4_10 sf .412(U)A f0_10 sf (.)S 59 378 :M -.004(Lemma 13 guarantees that this recursive definition is sound, because it guarantees that if )A 417 378 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 428 378 :M .694 .069(is )J 439 378 :M .056 .006(a )J 447 378 :M .688 .069(minimal )J 485 378 :M -1.328(d-)A 59 390 :M -.037(connecting )A 106 390 :M .202 .02(path )J 127 390 :M -.116(between )A 163 390 :M .255 .026(X )J 174 390 :M -.313(and )A 191 390 :M .255 .026(Y )J 202 390 :M .189 .019(given )J 228 390 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 239 390 :M .361 .036(that )J 258 390 :M .098 .01(contains )J 295 390 :M -.286(hidden )A 324 390 :M -.009(variables, )A 366 390 :M .202 .02(then )J 388 390 :M -.097(there )A 412 390 :M .694 .069(is )J 424 390 :M .056 .006(a )J 433 390 :M .454 .045(0 )J 443 390 :M -.319(order )A 467 390 :M -.443(hidden)A 59 402 :M -.029(variable on )A f4_10 sf -.054(U)A f0_10 sf -.029( and also that the definition of the order of any hidden variable W on )A f4_10 sf -.054(U)A f0_10 sf ( )S 399 402 :M .694 .069(is )J 410 402 :M .555 .056(not )J 427 402 :M -.355(defined )A 458 402 :M .601 .06(in )J 470 402 :M -.053(terms)A 59 414 :M -.053(of the order of W.)A 59 429 :M f2_10 sf .232(Lemma)A f2_12 sf .096 .01( )J f2_10 sf .157(14:)A f0_10 sf .344 .034( If there is an edge A *)J f1_10 sf .349A f0_10 sf .43 .043( B in MAG\()J f4_10 sf .255(G)A f0_10 sf .118<28>A f2_10 sf .275(O)A f0_10 sf .088(,)A f2_10 sf .197(S)A f0_10 sf .088(,)A f2_10 sf .236(L)A f0_10 sf .351 .035(\)\), then in )J f4_10 sf .255(G)A f0_10 sf .118<28>A f2_10 sf .275(O)A f0_10 sf .088(,)A f2_10 sf .197(S)A f0_10 sf .088(,)A f2_10 sf .236(L)A f0_10 sf .371 .037(\) there )J 412 429 :M .694 .069(is )J 423 429 :M .051 .005(an )J 436 429 :M -.062(inducing )A 474 429 :M -.072(path)A 59 441 :M .047 .005(between A and B that is into B.)J 59 456 :M f2_10 sf .78(Proof.)A f0_10 sf .444 .044( )J 94 456 :M -.085(By )A 109 456 :M .218 .022(the )J 126 456 :M -.038(definition )A 169 456 :M .144 .014(of )J 182 456 :M .056 .006(a )J 191 456 :M .132 .013(MAG )J 219 456 :M -.313(and )A 237 456 :M (Lemma )S 272 456 :M .454 .045(5 )J 282 456 :M -.097(there )A 306 456 :M .694 .069(is )J 318 456 :M .051 .005(an )J 332 456 :M -.062(inducing )A 371 456 :M .202 .02(path )J 393 456 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 405 456 :M -.116(between )A 442 456 :M .255 .026(A )J 454 456 :M -.313(and )A 472 456 :M -.17(B )A 483 456 :M .222(in)A 59 468 :M f4_10 sf .246(G)A f0_10 sf .113<28>A f2_10 sf .265(O)A f0_10 sf .085(,)A f2_10 sf .189(S)A f0_10 sf .085(,)A f2_10 sf .227(L)A f0_10 sf .309 .031(\), and B is not an )J 170 468 :M -.101(ancestor )A 206 468 :M .144 .014(of )J 218 468 :M .255 .026(A )J 229 468 :M .144 .014(or )J 241 468 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 252 468 :M .601 .06(in )J 264 468 :M f4_10 sf .513(G)A f0_10 sf .236<28>A f2_10 sf .553(O)A f0_10 sf .178(,)A f2_10 sf .395(S)A f0_10 sf .178(,)A f2_10 sf .474(L)A f0_10 sf .493 .049(\). )J 312 468 :M .361 .036(Suppose )J 350 468 :M .361 .036(that )J 369 468 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 380 468 :M .694 .069(is )J 391 468 :M .555 .056(out )J 408 468 :M .144 .014(of )J 420 468 :M .275 .028(B. )J 433 468 :M -.078(If )A 443 468 :M -.097(there )A 466 468 :M -.235(are )A 481 468 :M (no)S 59 480 :M .068 .007(colliders on )J f4_10 sf (U)S f0_10 sf .041 .004(, then )J f4_10 sf (U)S f0_10 sf .049 .005( is a directed path from B to A, and B is an ancestor of A, which is )J 419 480 :M .056 .006(a )J 427 480 :M -.054(contradiction. )A 485 480 :M -.655(If)A 59 492 :M .105 .01(there is a collider C on )J f4_10 sf .069(U)A f0_10 sf .11 .011( closest )J 194 492 :M .601 .06(to )J 206 492 :M .275 .028(B, )J 219 492 :M -.313(and )A 236 492 :M .786 .079(it )J 246 492 :M .694 .069(is )J 257 492 :M .051 .005(an )J 270 492 :M -.101(ancestor )A 306 492 :M .144 .014(of )J 318 492 :M .275 .028(B, )J 331 492 :M .65 .065(A, )J 345 492 :M .144 .014(or )J 357 492 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(. )J 371 492 :M -.078(If )A 381 492 :M .755 .075(C )J 392 492 :M .694 .069(is )J 403 492 :M .051 .005(an )J 416 492 :M -.101(ancestor )A 452 492 :M .144 .014(of )J 464 492 :M -.17(B )A 474 492 :M -.072(then)A 59 504 :M .201 .02(there is a cycle in )J f4_10 sf .14(G)A f0_10 sf .065<28>A f2_10 sf .151(O)A f0_10 sf (,)S f2_10 sf .108(S)A f0_10 sf (,)S f2_10 sf .129(L)A f0_10 sf .221 .022(\) which is a contradiction. If C is an )J 319 504 :M -.101(ancestor )A 355 504 :M .144 .014(of )J 367 504 :M .255 .026(A )J 378 504 :M .144 .014(or )J 390 504 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(, )J 404 504 :M .202 .02(then )J 425 504 :M -.17(B )A 435 504 :M .694 .069(is )J 446 504 :M .051 .005(an )J 459 504 :M -.187(ancestor)A 59 516 :M .091 .009(of A or )J f2_10 sf .061(S)A f0_10 sf .156 .016( which is a contradiction. Hence )J f4_10 sf .08(U)A f0_10 sf .088 .009( is into B. )J f1_10 sf <5C>S 59 531 :M f2_10 sf 2.491 .249(Lemma )J 101 531 :M .89(15:)A f0_10 sf .501 .05( )J 123 531 :M -.078(If )A 135 531 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 214 531 :M -.313(and )A 233 531 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 320 531 :M -.094(have )A 344 531 :M .218 .022(the )J 362 531 :M (same )S 389 531 :M (basic )S 416 531 :M -.043(colliders, )A 459 531 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 473 531 :M .694 .069(is )J 487 531 :M (a)S 59 543 :M -.019(discriminating )A 120 543 :M .202 .02(path )J 141 543 :M -.116(between )A 177 543 :M .159(X)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .055 .006( )J 191 543 :M -.313(and )A 208 543 :M .255 .026(Y )J 219 543 :M -.052(for )A 234 543 :M .855 .086(F )J 244 543 :M .601 .06(in )J 256 543 :M .185(MAG\()A f4_10 sf .201(G)A f0_6 sf 0 2 rm .083(1)A 0 -2 rm f0_10 sf .092<28>A f2_10 sf .216(O)A f0_10 sf .069(,)A f2_10 sf .154(S)A f0_10 sf .069(,)A f2_10 sf .185(L)A f0_10 sf .249 .025(\)\), )J 336 543 :M f4_10 sf <55D5>S f0_10 sf ( )S 350 543 :M .694 .069(is )J 361 543 :M .218 .022(the )J 377 543 :M .202 .02(path )J 398 543 :M -.131(corresponding )A 458 543 :M .601 .06(to )J 471 543 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 483 543 :M .222(in)A 59 555 :M .016(MAG\()A f4_10 sf (G)S f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf <53D5>S f0_10 sf (,)S f2_10 sf <4CD5>S f0_10 sf .032 .003(\)\), and every vertex \(except for the endpoints and possibly F\) is a collider )J 429 555 :M .417 .042(on )J 443 555 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(, )J 460 555 :M .202 .02(then )J 481 555 :M f4_10 sf -.547<55D5>A 59 567 :M f0_10 sf .37 .037(is a discriminating path for F in MAG\()J f4_10 sf .186(G)A f0_6 sf 0 2 rm .077(2)A 0 -2 rm f0_10 sf .086<28>A f2_10 sf .2(O)A f0_10 sf .064(,)A f2_10 sf .115<53D5>A f0_10 sf .064(,)A f2_10 sf .129<4CD5>A f0_10 sf .118(\)\).)A 59 582 :M f2_10 sf .11(Proof.)A f0_10 sf .325 .032( Suppose that the vertices on )J f4_10 sf .18(U)A f0_10 sf .32 .032( preceding F are X)J f0_6 sf 0 2 rm .075(1)A 0 -2 rm f0_10 sf .113 .011(, )J 299 582 :M 1.786 .179(..., )J 315 582 :M .37(X)A f0_6 sf 0 2 rm .154(n)A 0 -2 rm f0_10 sf .233 .023(. )J 332 582 :M -.085(By )A 347 582 :M .057 .006(definition, )J 392 582 :M .601 .06(in )J 404 582 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 481 582 :M -.155(X)A f0_6 sf 0 2 rm (1)S 0 -2 rm 59 594 :M f0_10 sf .694 .069(is )J 70 594 :M .555 .056(not )J 87 594 :M -.226(adjacent )A 122 594 :M .601 .06(to )J 134 594 :M .65 .065(Y, )J 148 594 :M .159(X)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 162 594 :M .694 .069(is )J 173 594 :M .056 .006(a )J 181 594 :M -.13(collider )A 214 594 :M .417 .042(on )J 228 594 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 242 594 :M -.313(and )A 259 594 :M .159(X)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 273 594 :M .694 .069(is )J 284 594 :M .051 .005(an )J 297 594 :M -.182(unshielded )A 342 594 :M -.114(non-collider )A 393 594 :M .417 .042(on )J 407 594 :M .218 .022(the )J 424 594 :M -.079(concatenation )A 483 594 :M -.328(of)A 59 606 :M f4_10 sf .096(U)A f0_10 sf .07(\(X)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .065(,X)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .157 .016(\) and the edge X)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf ( )S f1_10 sf .131A f0_10 sf .186 .019( Y. Hence in MAG\()J f4_10 sf .096(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .103(O)A f0_10 sf (,)S f2_10 sf .059<53D5>A f0_10 sf (,)S f2_10 sf .066<4CD5>A f0_10 sf .167 .017(\)\), X)J f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .067 .007( is )J 337 606 :M .555 .056(not )J 354 606 :M -.226(adjacent )A 389 606 :M .601 .06(to )J 401 606 :M .65 .065(Y, )J 415 606 :M .159(X)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 429 606 :M .694 .069(is )J 440 606 :M .056 .006(a )J 448 606 :M -.13(collider )A 481 606 :M (on)S 59 618 :M f4_10 sf -.052<55D5>A f0_10 sf -.041(, and X)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf -.04( is an unshielded non-collider on the concatenation of )A 317 618 :M f4_10 sf -.079<55D5>A f0_10 sf -.079(\(X)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.073(,X)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf -.088(\) )A 360 618 :M -.313(and )A 377 618 :M .218 .022(the )J 393 618 :M -.344(edge )A 414 618 :M -.116(between )A 450 618 :M .159(X)A f0_6 sf 0 2 rm .066(2)A 0 -2 rm f0_10 sf .055 .006( )J 464 618 :M -.313(and )A 481 618 :M .281(Y.)A 59 630 :M .157 .016(It follows that the edge between X)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .118 .012( and Y is oriented as X)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf ( )S f1_10 sf .106A f0_10 sf .132 .013( Y in MAG\()J f4_10 sf .077(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .083(O)A f0_10 sf (,)S f2_10 sf .048<53D5>A f0_10 sf (,)S f2_10 sf .054<4CD5>A f0_10 sf .049(\)\).)A 59 645 :M .715 .072(Suppose for each X)J f0_6 sf 0 2 rm .073(i)A 0 -2 rm f0_10 sf .297 .03(, 2 )J cF f1_10 sf .03A sf .297 .03( i )J cF f1_10 sf .03A sf .297 .03( m)J f1_10 sf .24(-)A f0_10 sf .683 .068(1, in MAG\()J f4_10 sf .315(G)A f0_6 sf 0 2 rm .131(2)A 0 -2 rm f0_10 sf .145<28>A f2_10 sf .34(O)A f0_10 sf .109(,)A f2_10 sf .194<53D5>A f0_10 sf .109(,)A f2_10 sf .218<4CD5>A f0_10 sf .511 .051(\)\) X)J f0_6 sf 0 2 rm .073(i)A 0 -2 rm f0_10 sf .222 .022( is )J 320 645 :M .056 .006(a )J 328 645 :M -.13(collider )A 361 645 :M .417 .042(on )J 375 645 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(, )J 392 645 :M -.313(and )A 409 645 :M .218 .022(the )J 425 645 :M -.344(edge )A 446 645 :M -.116(between )A 482 645 :M .092(X)A f0_6 sf 0 2 rm (i)S 0 -2 rm 59 657 :M f0_10 sf .046 .005(and Y is )J 96 657 :M -.158(oriented )A 131 657 :M .144 .014(as )J 143 657 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 156 657 :M f1_10 sf .504A f0_10 sf .128 .013( )J 170 657 :M .65 .065(Y. )J 184 657 :M .144 .014(In )J 196 657 :M .278(MAG\()A f4_10 sf .301(G)A f0_6 sf 0 2 rm .125(2)A 0 -2 rm f0_10 sf .139<28>A f2_10 sf .324(O)A f0_10 sf .104(,)A f2_10 sf .185<53D5>A f0_10 sf .104(,)A f2_10 sf .208<4CD5>A f0_10 sf .374 .037(\)\), )J 284 657 :M .388 .039(let )J 298 657 :M f4_10 sf .421<56D5>A f0_10 sf .223 .022( )J 312 657 :M .051 .005(be )J 325 657 :M .218 .022(the )J 341 657 :M -.079(concatenation )A 399 657 :M .144 .014(of )J 411 657 :M f4_10 sf -.063<55D5>A f0_10 sf -.063(\(X)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.058(,X)A f0_6 sf 0 2 rm -.055(m)A 0 -2 rm f1_6 sf 0 2 rm -.037(-1)A 0 -2 rm f0_10 sf -.069(\) )A 462 657 :M -.313(and )A 479 657 :M -.108(the)A 59 669 :M -.022(edge between X)A f0_6 sf 0 2 rm (m-1)S 0 -2 rm f0_10 sf -.02( and Y. Every vertex on )A f4_10 sf <56D5>S f0_10 sf -.022( between X)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.021( and X)A f0_6 sf 0 2 rm (m)S 0 -2 rm f0_10 sf -.019( is a collider by hypothesis, and for each X)A f0_6 sf 0 2 rm (i)S 0 -2 rm 59 681 :M f0_10 sf .172 .017(between X)J f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .078 .008( and X)J f0_6 sf 0 2 rm (m)S 0 -2 rm f0_10 sf .084 .008(, there is an edge X)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( )S f1_10 sf .075A f0_10 sf .103 .01( Y by hypothesis. Hence )J f4_10 sf .036<56D5>A f0_10 sf .1 .01( is a discriminating path )J 440 681 :M -.052(for )A 455 681 :M .476(X)A f0_6 sf 0 2 rm .308(m)A 0 -2 rm f0_10 sf .3 .03(. )J 474 681 :M -.078(If )A 484 681 :M f4_10 sf (V)S endp %%Page: 8 8 %%BeginPageSetup initializepage (peter; page: 8 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (8)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf .281 .028(is the corresponding path in MAG\()J f4_10 sf .12(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .055<28>A f2_10 sf .129(O)A f0_10 sf (,)S f2_10 sf .092(S)A f0_10 sf (,)S f2_10 sf .111(L)A f0_10 sf .138 .014(\)\), )J f4_10 sf .102(V)A f0_10 sf .218 .022( is a discriminating )J 340 51 :M .522 .052(path, )J 364 51 :M -.313(and )A 381 51 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 397 51 :M .694 .069(is )J 408 51 :M .056 .006(a )J 416 51 :M -.114(non-collider )A 467 51 :M .417 .042(on )J 481 51 :M f4_10 sf .987(V)A f0_10 sf (.)S 59 63 :M .177 .018(Hence X)J f0_6 sf 0 2 rm (m)S 0 -2 rm f0_10 sf .098 .01( is a non-collider on )J f4_10 sf .04<56D5>A f0_10 sf (. )S 198 63 :M .328 .033(It )J 208 63 :M .298 .03(follows )J 242 63 :M .361 .036(that )J 261 63 :M .218 .022(the )J 277 63 :M -.344(edge )A 298 63 :M -.116(between )A 334 63 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 350 63 :M -.313(and )A 367 63 :M .255 .026(Y )J 378 63 :M .694 .069(is )J 389 63 :M -.158(oriented )A 424 63 :M .144 .014(as )J 436 63 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 452 63 :M f1_10 sf .504A f0_10 sf .128 .013( )J 466 63 :M .65 .065(Y. )J 480 63 :M -.67(By)A 59 75 :M .328 .033(induction, )J f4_10 sf .085<55D5>A f0_10 sf .213 .021( is a discriminating path for F in MAG\()J f4_10 sf .116(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .054<28>A f2_10 sf .125(O)A f0_10 sf (,)S f2_10 sf .072<53D5>A f0_10 sf (,)S f2_10 sf .08<4CD5>A f0_10 sf .134 .013(\)\). )J f1_10 sf <5C>S 59 90 :M f2_10 sf .984 .098(Lemma 16:)J f0_10 sf .526 .053( If MAG\()J f4_10 sf .278(G)A f0_6 sf 0 2 rm .115(1)A 0 -2 rm f0_10 sf .128<28>A f2_10 sf .299(O)A f0_10 sf .096(,)A f2_10 sf .214(S)A f0_10 sf .096(,)A f2_10 sf .257(L)A f0_10 sf .654 .065(\)\) and MAG\()J f4_10 sf .278(G)A f0_6 sf 0 2 rm .115(2)A 0 -2 rm f0_10 sf .128<28>A f2_10 sf .299(O)A f0_10 sf .096(,)A f2_10 sf .171<53D5>A f0_10 sf .096(,)A f2_10 sf .192<4CD5>A f0_10 sf .424 .042(\)\) have the )J 339 90 :M (same )S 363 90 :M (basic )S 387 90 :M -.043(colliders, )A 427 90 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 438 90 :M .694 .069(is )J 449 90 :M .056 .006(a )J 457 90 :M .112(minimal)A 59 102 :M -.142(d-connecting )A 114 102 :M .202 .02(path )J 136 102 :M -.116(between )A 173 102 :M .255 .026(X )J 185 102 :M -.313(and )A 203 102 :M .255 .026(Y )J 215 102 :M .189 .019(given )J 243 102 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 256 102 :M .601 .06(in )J 270 102 :M .185(MAG\()A f4_10 sf .201(G)A f0_6 sf 0 2 rm .083(1)A 0 -2 rm f0_10 sf .092<28>A f2_10 sf .216(O)A f0_10 sf .069(,)A f2_10 sf .154(S)A f0_10 sf .069(,)A f2_10 sf .185(L)A f0_10 sf .249 .025(\)\), )J 352 102 :M f4_10 sf <55D5>S f0_10 sf ( )S 368 102 :M .694 .069(is )J 381 102 :M .218 .022(the )J 399 102 :M -.131(corresponding )A 460 102 :M .202 .02(path )J 483 102 :M .222(in)A 59 114 :M .149(MAG\()A f4_10 sf .162(G)A f0_6 sf 0 2 rm .067(2)A 0 -2 rm f0_10 sf .074<28>A f2_10 sf .174(O)A f0_10 sf .056(,)A f2_10 sf .099<53D5>A f0_10 sf .056(,)A f2_10 sf .112<4CD5>A f0_10 sf .231 .023(\)\), then F is a collider on )J f4_10 sf .162(U)A f0_10 sf .21 .021( if and only if F is a collider on )J f4_10 sf .118<55D5>A f0_10 sf (.)S 59 129 :M f2_10 sf .113(Proof.)A f0_10 sf .254 .025( If F is not a hidden vertex on )J f4_10 sf .185(U)A f0_10 sf .448 .045(, then because MAG\()J f4_10 sf .185(G)A f0_6 sf 0 2 rm .077(1)A 0 -2 rm f0_10 sf .085<28>A f2_10 sf .2(O)A f0_10 sf .064(,)A f2_10 sf .143(S)A f0_10 sf .064(,)A f2_10 sf .171(L)A f0_10 sf .196 .02(\)\) )J 355 129 :M -.313(and )A 372 129 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 457 129 :M -.094(have )A 479 129 :M -.108(the)A 59 141 :M .076 .008(same basic colliders, F is not a hidden vertex on )J f4_10 sf .031<55D5>A f0_10 sf .065 .006(, and by definition F is )J 360 141 :M .056 .006(a )J 368 141 :M -.13(collider )A 401 141 :M .417 .042(on )J 415 141 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 426 141 :M .328 .033(if )J 436 141 :M -.313(and )A 453 141 :M .515 .052(only )J 475 141 :M .328 .033(if )J 485 141 :M (F)S 59 153 :M .045 .004(is a collider on )J f4_10 sf <55D5>S f0_10 sf (.)S 59 168 :M .186 .019(Suppose F is a hidden vertex on )J f4_10 sf .105(U)A f0_10 sf .223 .022(. By Lemma 8, MAG\()J f4_10 sf .105(G)A f0_10 sf <28>S f2_10 sf .113(O)A f0_10 sf (,)S f2_10 sf .081(S)A f0_10 sf (,)S f2_10 sf .097(L)A f0_10 sf .111 .011(\)\) )J 335 168 :M .098 .01(contains )J 372 168 :M .047 .005(one )J 390 168 :M .144 .014(of )J 402 168 :M .218 .022(the )J 418 168 :M (subgraphs )S 462 168 :M .144 .014(of )J 474 168 :M -.072(type)A 59 180 :M .186 .019(\(i\) through \(viii\) in Figure 1. By Lemma 9, )J f4_10 sf .104(U)A f0_10 sf .219 .022( contains a discriminating path )J f4_10 sf .104(U)A f0_10 sf .231 .023(\(M,N\) for F.)J 59 195 :M .091 .009(Suppose first that F is a zero order )J 201 195 :M -.286(hidden )A 230 195 :M -.081(vertex )A 258 195 :M .417 .042(on )J 272 195 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 286 195 :M (Then )S 310 195 :M .388 .039(all )J 324 195 :M .144 .014(of )J 336 195 :M .218 .022(the )J 352 195 :M -.074(vertices )A 386 195 :M .417 .042(on )J 400 195 :M f4_10 sf .147(U)A f0_10 sf .354 .035(\(M,F\) )J 435 195 :M -.099(except )A 464 195 :M -.052(for )A 479 195 :M -.108(the)A 59 207 :M .017 .002(endpoints are unshielded colliders in MAG\()J f4_10 sf (G)S f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf .018 .002(\)\). Because MAG\()J f4_10 sf (G)S f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf (\)\) and MAG\()S f4_10 sf (G)S f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf <53D5>S f0_10 sf (,)S f2_10 sf <4CD5>S f0_10 sf <2929>S 59 219 :M .038 .004(have the same basic colliders, all of the vertices on )J f4_10 sf (U)S f0_10 sf .031 .003(\(M,F\) are )J 317 219 :M -.182(unshielded )A 362 219 :M -.043(colliders. )A 402 219 :M -.207(Hence )A 430 219 :M .417 .042(by )J 444 219 :M (Lemma )S 478 219 :M .25(15,)A 59 231 :M f4_10 sf .1<55D5>A f0_10 sf .306 .031(\(A,B\) is a discriminating path in MAG\()J f4_10 sf .136(G)A f0_6 sf 0 2 rm .057(2)A 0 -2 rm f0_10 sf .063<28>A f2_10 sf .147(O)A f0_10 sf (,)S f2_10 sf .084<53D5>A f0_10 sf (,)S f2_10 sf .094<4CD5>A f0_10 sf .205 .021(\)\). It follows that F is a collider on )J f4_10 sf .136(U)A f0_10 sf .179 .018( if and only if F)J 59 243 :M .045 .004(is a collider on )J f4_10 sf <55D5>S f0_10 sf (.)S 59 258 :M .349 .035(Suppose that )J 115 258 :M -.052(for )A 130 258 :M .388 .039(all )J 144 258 :M .454 .045(0 )J 153 258 :M cF f1_10 sf .092A sf .92 .092( )J 163 258 :M .656 .066(i )J 170 258 :M .782 .078(< )J 180 258 :M .833 .083(n, )J 192 258 :M .218 .022(the )J 208 258 :M .295(i)A f0_6 sf 0 -4 rm .247(th)A 0 4 rm f0_10 sf .265 .027( )J 220 258 :M -.319(order )A 243 258 :M -.286(hidden )A 272 258 :M -.074(vertices )A 306 258 :M .417 .042(on )J 320 258 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 331 258 :M -.235(are )A 346 258 :M -.158(oriented )A 381 258 :M .218 .022(the )J 397 258 :M (same )S 421 258 :M -.053(way )A 441 258 :M .417 .042(on )J 455 258 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(. )J 472 258 :M -.219(Now)A 59 270 :M -.016(consider an n)A f0_6 sf 0 -4 rm (th)S 0 4 rm f0_10 sf -.015( order hidden vertex on )A f4_10 sf (U)S f0_10 sf -.015(. There is a subpath )A f4_10 sf (U)S f0_10 sf -.016(\(M,N\) that is )A 362 270 :M .056 .006(a )J 370 270 :M -.019(discriminating )A 431 270 :M .202 .02(path )J 452 270 :M -.052(for )A 467 270 :M 1.201 .12(F. )J 480 270 :M -.67(By)A 59 282 :M (the induction )S 115 282 :M .584 .058(hypothesis, )J 165 282 :M .388 .039(all )J 179 282 :M .144 .014(of )J 191 282 :M .218 .022(the )J 207 282 :M -.103(colliders )A 244 282 :M .417 .042(on )J 258 282 :M f4_10 sf (U)S f0_10 sf (\(M,N\) )S 294 282 :M -.235(are )A 309 282 :M -.103(colliders )A 346 282 :M .417 .042(on )J 360 282 :M f4_10 sf <55D5>S f0_10 sf .088 .009(\(M,N\). )J 402 282 :M -.207(Hence )A 430 282 :M .417 .042(by )J 444 282 :M (Lemma )S 478 282 :M .25(15,)A 59 294 :M f4_10 sf .1<55D5>A f0_10 sf .306 .031(\(A,B\) is a discriminating path in MAG\()J f4_10 sf .136(G)A f0_6 sf 0 2 rm .057(2)A 0 -2 rm f0_10 sf .063<28>A f2_10 sf .147(O)A f0_10 sf (,)S f2_10 sf .084<53D5>A f0_10 sf (,)S f2_10 sf .094<4CD5>A f0_10 sf .205 .021(\)\). It follows that F is a collider on )J f4_10 sf .136(U)A f0_10 sf .179 .018( if and only if F)J 59 306 :M -.013(is a collider on )A f4_10 sf <55D5>S f0_10 sf (. )S f1_10 sf <5C>S 59 321 :M f2_10 sf .559 .056(Lemma 17:)J f0_10 sf .265 .026( If A and B are d-connected given )J f2_10 sf .158(R)A f0_10 sf .308 .031( in MAG\()J f4_10 sf .158(G)A f0_10 sf .073<28>A f2_10 sf .17(O)A f0_10 sf .055(,)A f2_10 sf .121(S)A f0_10 sf .055(,)A f2_10 sf .146(L)A f0_10 sf .245 .024(\)\), then )J 367 321 :M .255 .026(A )J 378 321 :M -.313(and )A 395 321 :M -.17(B )A 405 321 :M -.235(are )A 420 321 :M -.305(d-connected )A 469 321 :M -.054(given)A 59 333 :M f2_10 sf .67(R)A f0_10 sf .211 .021( )J f1_10 sf .712A f0_10 sf .211 .021( )J f2_10 sf .516(S)A f0_10 sf .494 .049( in )J f4_10 sf .67(G)A f0_10 sf .309<28>A f2_10 sf .721(O)A f0_10 sf .232(,)A f2_10 sf .516(S)A f0_10 sf .232(,)A f2_10 sf .619(L)A f0_10 sf .54(\).)A 59 348 :M f2_10 sf .78(Proof.)A f0_10 sf .444 .044( )J 94 348 :M .361 .036(Suppose )J 132 348 :M .361 .036(that )J 151 348 :M .601 .06(in )J 163 348 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 237 348 :M .255 .026(A )J 248 348 :M -.313(and )A 265 348 :M -.17(B )A 275 348 :M -.235(are )A 290 348 :M -.305(d-connected )A 339 348 :M .189 .019(given )J 365 348 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 378 348 :M .417 .042(by )J 393 348 :M .056 .006(a )J 402 348 :M .688 .069(minimal )J 441 348 :M -.2(d-connecting)A 59 360 :M .184 .018(path )J f4_10 sf .101(U)A f0_10 sf .173 .017(. Then each vertex on )J f4_10 sf .101(U)A f0_10 sf .154 .015( is active given )J f2_10 sf .101(R)A f0_10 sf .159 .016(. For each edge X *)J f1_10 sf .14A f0_10 sf .185 .019(* Y in MAG\()J f4_10 sf .101(G)A f0_10 sf <28>S f2_10 sf .109(O)A f0_10 sf (,)S f2_10 sf .078(S)A f0_10 sf (,)S f2_10 sf .094(L)A f0_10 sf .126 .013(\)\), )J 448 360 :M -.097(there )A 471 360 :M .694 .069(is )J 482 360 :M -.439(an)A 59 372 :M .129 .013(inducing path between X and Y in )J f4_10 sf .069(G)A f0_10 sf <28>S f2_10 sf .074(O)A f0_10 sf (,)S f2_10 sf .053(S)A f0_10 sf (,)S f2_10 sf .064(L)A f0_10 sf .121 .012(\). By Lemma 2, Lemma 3, and Lemma 4, there is a path that d-)J 59 384 :M -.081(connects X and )A 123 384 :M .255 .026(Y )J 134 384 :M .189 .019(given )J 160 384 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 172 384 :M f1_10 sf .62A f0_10 sf .202 .02( )J 184 384 :M f2_10 sf .093(S)A f0_10 sf .332 .033(\\{X,Y} )J 223 384 :M .601 .06(in )J 235 384 :M f4_10 sf .513(G)A f0_10 sf .236<28>A f2_10 sf .553(O)A f0_10 sf .178(,)A f2_10 sf .395(S)A f0_10 sf .178(,)A f2_10 sf .474(L)A f0_10 sf .493 .049(\). )J 283 384 :M .314 .031(Choose )J 317 384 :M .388 .039(all )J 331 384 :M .123 .012(such )J 353 384 :M -.142(d-connecting )A 407 384 :M .202 .02(path )J 428 384 :M -.052(for )A 443 384 :M -.204(each )A 464 384 :M (pair )S 483 384 :M -.328(of)A 59 396 :M .025 .002(vertices X and Y adjacent on )J f4_10 sf (U)S f0_10 sf .027 .003(; call this collection of d-connecting paths )J f3_10 sf (T)S f1_10 sf ( )S f0_10 sf .016 .002(. If a vertex X is on )J 445 396 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 459 396 :M .133 .013(say )J 476 396 :M (that)S 59 408 :M .448 .045(X is )J f2_10 sf .757 .076(active in )J f3_10 sf .362(T)A f0_10 sf .135 .013( )J f2_10 sf .27(given)A f0_10 sf .135 .013( )J f2_10 sf .428(R)A f0_10 sf .135 .013( )J f1_10 sf .455A f0_10 sf .135 .013( )J f2_10 sf .329(S)A f0_10 sf .789 .079( whenever either \(i\) there are vertices C )J 354 408 :M -.313(and )A 371 408 :M .255 .026(D )J 382 408 :M .417 .042(on )J 396 408 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 407 408 :M -.226(adjacent )A 442 408 :M .5 .05( )J 446 408 :M .601 .06(to )J 458 408 :M .65 .065(X, )J 472 408 :M -.246(there)A 59 420 :M .161 .016(is a path in )J f3_10 sf .113(T)A f0_10 sf .193 .019( between X and C that is into X, there is a path in )J f3_10 sf .113(T)A f0_10 sf .199 .02( between X and D that is into X, and X )J 484 420 :M .332(is)A 59 432 :M .534 .053(an ancestor of )J f2_10 sf .296(R)A f0_10 sf .093 .009( )J f1_10 sf .315A f0_10 sf .093 .009( )J f2_10 sf .228(S)A f0_10 sf .218 .022( in )J f4_10 sf .296(G)A f0_10 sf .136<28>A f2_10 sf .319(O)A f0_10 sf .103(,)A f2_10 sf .228(S)A f0_10 sf .103(,)A f2_10 sf .274(L)A f0_10 sf .389 .039(\), or \(ii\) there is )J 266 432 :M .056 .006(a )J 274 432 :M .5 .05( )J 278 432 :M -.142(d-connecting )A 332 432 :M .202 .02(path )J 353 432 :M .601 .06(in )J 365 432 :M f3_10 sf .278(T)A f0_10 sf .114 .011( )J 375 432 :M .144 .014(containing )J 421 432 :M .255 .026(X )J 432 432 :M .144 .014(as )J 444 432 :M .051 .005(an )J 457 432 :M -.142(endpoint)A 59 444 :M .19 .019(that is out of X, and X is not in )J f2_10 sf .146(R)A f0_10 sf .342 .034(. Consider the following three cases.)J 59 459 :M f4_10 sf .156(U)A f0_10 sf .275 .027( contains a subpath C *)J f1_10 sf .216A f0_10 sf .177 .018(o F o)J f1_10 sf .216A f0_10 sf .213 .021(* D, and F is active on )J f4_10 sf .156(U)A f0_10 sf .217 .022( given )J f2_10 sf .156(R)A f0_10 sf .234 .023(. Hence there is a path )J f4_10 sf .132(X)A f0_6 sf 0 2 rm .065(1)A 0 -2 rm f0_10 sf .115 .011( in )J f3_10 sf .132(T)A f0_10 sf .226 .023( that d-)J 59 471 :M .248 .025(connects C and F given \()J f2_10 sf .133(R)A f0_10 sf ( )S f1_10 sf .142A f0_10 sf ( )S f2_10 sf .082(S\))A f0_10 sf .236 .024(\\{C,F}, and a path )J f4_10 sf .113(X)A f0_6 sf 0 2 rm .055(2)A 0 -2 rm f0_10 sf .098 .01( in )J f3_10 sf .113(T)A f0_10 sf .231 .023( that d-connects F and D given )J 424 471 :M .243<28>A f2_10 sf .527(R)A f0_10 sf .183 .018( )J 439 471 :M f1_10 sf .62A f0_10 sf .202 .02( )J 451 471 :M f2_10 sf .136(S)A f0_10 sf .117(\)\\{F,D}.)A 59 483 :M .312 .031(F is not in )J f2_10 sf .274(R)A f0_10 sf .419 .042( because F is active on )J f4_10 sf .274(U)A f0_10 sf .383 .038( given )J f2_10 sf .274(R)A f0_10 sf .372 .037(, and F is not a collider on )J f4_10 sf .274(U)A f0_10 sf .334 .033(. F is active in )J f3_10 sf .232(T)A f0_10 sf .383 .038( given )J f2_10 sf .274(R)A f0_10 sf .086 .009( )J f1_10 sf .291A f0_10 sf .086 .009( )J f2_10 sf (S)S 59 495 :M f0_10 sf .177 .018(if )J f4_10 sf .163(X)A f0_6 sf 0 2 rm .08(1)A 0 -2 rm f0_10 sf .207 .021( and )J f4_10 sf .163(X)A f0_6 sf 0 2 rm .08(2)A 0 -2 rm f0_10 sf .298 .03( collide at F because it is an ancestor of )J f2_10 sf .148(S)A f0_10 sf .142 .014( in )J f4_10 sf .193(G)A f0_10 sf .089<28>A f2_10 sf .207(O)A f0_10 sf .067(,)A f2_10 sf .148(S)A f0_10 sf .067(,)A f2_10 sf .178(L)A f0_10 sf .275 .027(\), and is active in )J 400 495 :M f3_10 sf .278(T)A f0_10 sf .114 .011( )J 410 495 :M .328 .033(if )J 420 495 :M f4_10 sf .732(X)A f0_6 sf 0 2 rm .36(1)A 0 -2 rm f0_10 sf .3 .03( )J 434 495 :M -.313(and )A 451 495 :M f4_10 sf .732(X)A f0_6 sf 0 2 rm .36(2)A 0 -2 rm f0_10 sf .3 .03( )J 465 495 :M -.25(do )A 478 495 :M .111(not)A 59 507 :M .565 .057(collide at F because it is not in )J f2_10 sf .374(R)A f0_10 sf .118 .012( )J f1_10 sf .398A f0_10 sf .118 .012( )J f2_10 sf .288(S)A f0_10 sf .455 .046(, so F is active in )J f3_10 sf .317(T)A f0_10 sf .522 .052( given )J f2_10 sf .374(R)A f0_10 sf .118 .012( )J f1_10 sf .398A f0_10 sf .118 .012( )J f2_10 sf .288(S)A f0_10 sf (.)S 59 522 :M f4_10 sf .207(U)A f0_10 sf .366 .037( contains a subpath C *)J f1_10 sf .284A f0_10 sf .132 .013( F )J f1_10 sf .284A f0_10 sf .284 .028(* D, and F is active on )J f4_10 sf .207(U)A f0_10 sf .29 .029( given )J f2_10 sf .207(R)A f0_10 sf .315 .032(. It follows )J 380 522 :M .361 .036(that )J 399 522 :M .601 .06(in )J 411 522 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 485 522 :M (F)S 59 534 :M .093 .009(has a )J 83 534 :M -.292(descendant )A 128 534 :M .601 .06(in )J 140 534 :M f2_10 sf 1.052(R)A f0_10 sf .662 .066(. )J 155 534 :M -.207(Hence )A 183 534 :M .855 .086(F )J 193 534 :M .133 .013(has )J 210 534 :M .056 .006(a )J 218 534 :M -.292(descendant )A 263 534 :M .601 .06(in )J 275 534 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 287 534 :M .601 .06(in )J 299 534 :M f4_10 sf .513(G)A f0_10 sf .236<28>A f2_10 sf .553(O)A f0_10 sf .178(,)A f2_10 sf .395(S)A f0_10 sf .178(,)A f2_10 sf .474(L)A f0_10 sf .493 .049(\). )J 347 534 :M -.085(By )A 362 534 :M -.099(Lemma)A f0_12 sf ( )S 396 534 :M f0_10 sf .417 .042(14 )J 410 534 :M -.097(there )A 433 534 :M .694 .069(is )J 444 534 :M .051 .005(an )J 457 534 :M -.142(inducing)A 59 546 :M .063 .006(path between C and F that is into F, and an inducing path between D and )J 355 546 :M .855 .086(F )J 365 546 :M .361 .036(that )J 384 546 :M .694 .069(is )J 395 546 :M .674 .067(into )J 415 546 :M 1.201 .12(F. )J 428 546 :M .328 .033(It )J 438 546 :M .298 .03(follows )J 472 546 :M -.145(from)A 59 558 :M .207 .021(Lemma 2 and Lemma 3 that there is a path )J f4_10 sf .103(X)A f0_6 sf 0 2 rm .05(1)A 0 -2 rm f0_10 sf .093 .009( in )J 259 558 :M .356 .036(T )J 269 558 :M .361 .036(that )J 288 558 :M -.181(d-connects )A 333 558 :M .755 .075(C )J 344 558 :M -.313(and )A 361 558 :M .855 .086(F )J 371 558 :M .189 .019(given )J 397 558 :M .243<28>A f2_10 sf .527(R)A f0_10 sf .183 .018( )J 412 558 :M f1_10 sf .62A f0_10 sf .202 .02( )J 424 558 :M f2_10 sf .218(S)A f0_10 sf .758 .076(\)\\{C,F} )J 465 558 :M .361 .036(that )J 484 558 :M .332(is)A 59 570 :M .418 .042(into F, and a path )J f4_10 sf .244(X)A f0_6 sf 0 2 rm .12(2)A 0 -2 rm f0_10 sf .311 .031( in T that )J 185 570 :M -.181(d-connects )A 230 570 :M .855 .086(F )J 240 570 :M -.313(and )A 257 570 :M .255 .026(D )J 268 570 :M .189 .019(given )J 294 570 :M .243<28>A f2_10 sf .527(R)A f0_10 sf .183 .018( )J 309 570 :M f1_10 sf .62A f0_10 sf .202 .02( )J 321 570 :M f2_10 sf .136(S)A f0_10 sf .483 .048(\)\\{F,D} )J 362 570 :M .361 .036(that )J 381 570 :M .694 .069(is )J 392 570 :M .674 .067(into )J 412 570 :M 1.201 .12(F. )J 425 570 :M .855 .086(F )J 435 570 :M .694 .069(is )J 446 570 :M -.062(active )A 473 570 :M .601 .06(in )J 485 570 :M f3_10 sf (T)S 59 582 :M f0_10 sf .576 .058(given )J f2_10 sf .269(R)A f0_10 sf .085 .008( )J f1_10 sf .286A f0_10 sf .085 .008( )J f2_10 sf .207(S)A f0_10 sf .471 .047( because )J f4_10 sf .228(X)A f0_6 sf 0 2 rm .112(1)A 0 -2 rm f0_10 sf .29 .029( and )J f4_10 sf .228(X)A f0_6 sf 0 2 rm .112(2)A 0 -2 rm f0_10 sf .335 .034( collide at F in )J f4_10 sf .269(G)A f0_10 sf .124<28>A f2_10 sf .29(O)A f0_10 sf .093(,)A f2_10 sf .207(S)A f0_10 sf .093(,)A f2_10 sf .249(L)A f0_10 sf .388 .039(\), and F is an ancestor of )J f2_10 sf .269(R)A f0_10 sf .198 .02( in )J f4_10 sf .269(G)A f0_10 sf .124<28>A f2_10 sf .29(O)A f0_10 sf .093(,)A f2_10 sf .207(S)A f0_10 sf .093(,)A f2_10 sf .249(L)A f0_10 sf .217(\).)A 59 597 :M f4_10 sf .126(U)A f0_10 sf .222 .022( contains a subpath C *)J f1_10 sf .175A f0_10 sf .113 .011(* F )J f1_10 sf .172A f0_10 sf .165 .016( D, and F is active on )J f4_10 sf .126(U)A f0_10 sf .176 .018( given )J f2_10 sf .126(R)A f0_10 sf .224 .022(. \(The case where )J 404 597 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 415 597 :M .098 .01(contains )J 452 597 :M .056 .006(a )J 460 597 :M -.018(subpath)A 59 609 :M .181 .018(C )J f1_10 sf .234A f0_10 sf .154 .015( F *)J f1_10 sf .237A f0_10 sf .302 .03(* D is analogous.\) Because F is )J 237 609 :M -.062(active )A 264 609 :M .417 .042(on )J 278 609 :M f4_10 sf .951(U)A f2_10 sf .329 .033( )J 290 609 :M f0_10 sf .189 .019(given )J 316 609 :M f2_10 sf 1.052(R)A f0_10 sf .662 .066(, )J 331 609 :M .855 .086(F )J 341 609 :M .694 .069(is )J 352 609 :M .555 .056(not )J 369 609 :M .601 .06(in )J 381 609 :M f2_10 sf 1.052(R)A f0_10 sf .662 .066(. )J 396 609 :M -.163(There )A 422 609 :M .694 .069(is )J 433 609 :M .056 .006(a )J 441 609 :M -.337(directed )A 474 609 :M -.072(path)A 59 621 :M .225 .022(from F to D in )J f4_10 sf .175(G)A f0_10 sf .081<28>A f2_10 sf .188(O)A f0_10 sf .061(,)A f2_10 sf .135(S)A f0_10 sf .061(,)A f2_10 sf .162(L)A f0_10 sf .31 .031(\) that does not contain any vertices in )J 314 621 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(. )J 328 621 :M -.163(There )A 354 621 :M -.235(are )A 369 621 :M .387 .039(two )J 388 621 :M -.016(cases. )A 415 621 :M -.078(If )A 425 621 :M .218 .022(the )J 441 621 :M -.337(directed )A 474 621 :M -.072(path)A 59 633 :M .276 .028(contains a member of )J f2_10 sf .137(R)A f0_10 sf .196 .02(, then F is an ancestor of )J f2_10 sf .137(R)A f0_10 sf .199 .02(, and hence F is active in )J f3_10 sf .116(T)A f0_10 sf ( )S 380 633 :M .189 .019(given )J 406 633 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 418 633 :M f1_10 sf .62A f0_10 sf .202 .02( )J 430 633 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 441 633 :M -.203(regardless )A 483 633 :M -.328(of)A 59 645 :M -.024(whether or not the d-connecting paths collide at F. If the directed path does not contain )A 408 645 :M .056 .006(a )J 416 645 :M -.044(member )A 452 645 :M .144 .014(of )J 464 645 :M f2_10 sf 1.052(R)A f0_10 sf .662 .066(, )J 479 645 :M -.108(the)A 59 657 :M .17 .017(directed path d-connects F and D given )J f2_10 sf .084(R)A f0_10 sf ( )S f1_10 sf .089A f0_10 sf ( )S f2_10 sf .065(S)A f0_10 sf .095 .01( and is out of F. It )J 322 657 :M .298 .03(follows )J 356 657 :M .361 .036(that )J 375 657 :M .855 .086(F )J 385 657 :M .694 .069(is )J 396 657 :M -.062(active )A 423 657 :M .601 .06(in )J 435 657 :M f3_10 sf .278(T)A f0_10 sf .114 .011( )J 445 657 :M .189 .019(given )J 471 657 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 483 657 :M f1_10 sf S 59 669 :M f2_10 sf .13(S)A f0_10 sf .268 .027(. It follows from Lemma 1 that there is a path in )J f4_10 sf .169(G)A f0_10 sf .078<28>A f2_10 sf .182(O)A f0_10 sf .059(,)A f2_10 sf .13(S)A f0_10 sf .059(,)A f2_10 sf .156(L)A f0_10 sf .298 .03(\) that d-connects A and B given )J f2_10 sf .169(R)A f0_10 sf .053 .005( )J f1_10 sf .18A f0_10 sf .053 .005( )J f2_10 sf .13(S)A f0_10 sf .098 .01(. )J f1_10 sf <5C>S endp %%Page: 9 9 %%BeginPageSetup initializepage (peter; page: 9 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 485 713 6 12 rC 485 722 :M f0_12 sf (9)S gR gS 0 0 552 730 rC 59 51 :M f2_10 sf .171(Lemma)A f0_10 sf .059 .006( )J f2_10 sf .13(18)A f0_10 sf .321 .032(: If X and Y are d-connected given )J f2_10 sf .174(Z)A f0_10 sf .059 .006( )J f1_10 sf .2A f0_10 sf .059 .006( )J f2_10 sf .145(S)A f0_10 sf .139 .014( in )J f4_10 sf .188(G)A f0_10 sf .087<28>A f2_10 sf .203(O)A f0_10 sf .065(,)A f2_10 sf .145(S)A f0_10 sf .065(,)A f2_10 sf .174(L)A f0_10 sf .269 .027(\), then X and )J 383 51 :M .255 .026(Y )J 394 51 :M -.235(are )A 409 51 :M -.305(d-connected )A 458 51 :M .189 .019(given )J 484 51 :M f2_10 sf (Z)S 59 63 :M f0_10 sf .627 .063(in MAG\()J f4_10 sf .208(G)A f0_10 sf .096<28>A f2_10 sf .225(O)A f0_10 sf .072(,)A f2_10 sf .16(S)A f0_10 sf .072(,)A f2_10 sf .193(L)A f0_10 sf .132(\)\).)A 59 78 :M f2_10 sf .089(Proof.)A f0_10 sf .26 .026( Suppose that )J f4_10 sf .146(U)A f0_10 sf .287 .029( is a minimal d-connecting path between X and )J 347 78 :M .255 .026(Y )J 358 78 :M .189 .019(given )J 384 78 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 395 78 :M f1_10 sf .62A f0_10 sf .202 .02( )J 407 78 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 418 78 :M .601 .06(in )J 430 78 :M f4_10 sf .513(G)A f0_10 sf .236<28>A f2_10 sf .553(O)A f0_10 sf .178(,)A f2_10 sf .395(S)A f0_10 sf .178(,)A f2_10 sf .474(L)A f0_10 sf .493 .049(\). )J 478 78 :M -.877(We)A 59 90 :M .676 .068(will )J 79 90 :M -.1(perform )A 114 90 :M .056 .006(a )J 122 90 :M -.044(series )A 148 90 :M .144 .014(of )J 160 90 :M -.029(operation )A 201 90 :M .043 .004(which )J 229 90 :M .28 .028(show )J 254 90 :M .216 .022(how )J 275 90 :M .601 .06(to )J 287 90 :M -.017(construct )A 327 90 :M .056 .006(a )J 335 90 :M .202 .02(path )J 356 90 :M f4_10 sf <55D5>S f0_10 sf ( )S 370 90 :M .601 .06(in )J 382 90 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 456 90 :M .043 .004(which )J 484 90 :M -1.328(d-)A 59 102 :M .112 .011(connects A and B given )J f2_10 sf .056(Z)A f0_10 sf .118 .012( in MAG\()J f4_10 sf .06(G)A f0_10 sf <28>S f2_10 sf .065(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .056(L)A f0_10 sf .123 .012(\)\). The operations are illustrated with Figure 3 \()J f2_10 sf .056(Z)A f0_10 sf .062 .006( = O)J f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .041(,O)A f0_6 sf 0 2 rm (3)S 0 -2 rm f0_10 sf (\).)S 146 108 258 79 rC gS .61 .614 scale 237.58 184.159 :M f0_12 sf (X)S gR gS .61 .614 scale 247.411 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 265.434 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 268.711 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 184.159 :M f0_12 sf (O)S gR gS .61 .614 scale 278.542 187.418 :M f0_7 sf (1)S gR gS .61 .614 scale 283.457 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 286.734 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 290.011 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 291.65 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 294.927 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 298.204 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 301.481 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 304.758 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 308.035 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 309.673 184.159 :M f0_12 sf (L)S gR gS .61 .614 scale 317.865 187.418 :M f0_7 sf (1)S gR gS .61 .614 scale 321.142 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 324.419 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 327.696 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 330.973 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 332.612 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 335.889 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 339.166 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 342.443 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 345.72 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 348.997 184.159 :M f0_12 sf (L)S gR gS .61 .614 scale 355.551 187.418 :M f0_7 sf (2)S gR gS .61 .614 scale 358.828 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 362.105 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 365.381 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 368.658 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 371.935 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 373.574 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 376.851 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 380.128 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 383.405 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 386.682 184.159 :M f0_12 sf (O)S gR gS .61 .614 scale 394.874 187.418 :M f0_7 sf (2)S gR gS .61 .614 scale 398.151 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 401.428 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 404.705 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 407.982 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 409.62 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 412.897 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 416.174 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 419.451 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 422.728 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 426.005 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 427.644 184.159 :M f0_12 sf (L)S gR gS .61 .614 scale 435.836 187.418 :M f0_7 sf (3)S gR gS .61 .614 scale 439.113 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 442.39 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 445.667 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 448.944 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 450.583 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 453.859 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 457.136 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 460.413 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 463.69 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 466.967 184.159 :M f0_12 sf (L)S gR gS .61 .614 scale 473.521 187.418 :M f0_7 sf (4)S gR gS .61 .614 scale 476.798 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 480.075 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 483.352 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 486.629 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 489.906 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 491.545 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 494.822 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 498.099 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 501.375 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 504.652 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 507.929 184.159 :M f0_12 sf (Y)S gR gS .61 .614 scale 514.483 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 517.76 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 521.037 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 524.314 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 527.591 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 530.868 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 532.507 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 535.784 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 539.061 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 542.338 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 545.614 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 548.891 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 550.53 184.159 :M f0_12 sf (A)S gR gS .61 .614 scale 558.722 184.159 :M f0_12 sf (:)S gR gS .61 .614 scale 561.999 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 565.276 184.159 :M f0_12 sf (X)S gR gS .61 .614 scale 573.469 184.159 :M f0_12 sf (,)S gR gS .61 .614 scale 578.384 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 581.661 184.159 :M f0_12 sf (O)S gR gS .61 .614 scale 589.853 187.418 :M f0_7 sf (1)S gR gS .61 .614 scale 593.13 184.159 :M f0_12 sf (,)S gR gS .61 .614 scale 598.046 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 599.684 184.159 :M f0_12 sf (O)S gR gS .61 .614 scale 607.877 187.418 :M f0_7 sf (2)S gR gS .61 .614 scale 612.792 184.159 :M f0_12 sf (,)S gR gS .61 .614 scale 616.069 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 619.346 184.159 :M f0_12 sf (L)S gR gS .61 .614 scale 625.9 187.418 :M f0_7 sf (4)S gR gS .61 .614 scale 630.816 184.159 :M f0_12 sf (,)S gR gS .61 .614 scale 634.092 184.159 :M f0_12 sf ( )S gR gS .61 .614 scale 637.369 184.159 :M f0_12 sf (Y)S gR gS .61 .614 scale 237.58 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 240.857 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 244.134 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 247.411 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 265.434 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 268.711 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 273.626 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 276.903 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 280.18 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 283.457 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 286.734 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 290.011 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 291.65 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 294.927 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 298.204 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 301.481 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 304.758 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 308.035 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 309.673 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 312.95 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 316.227 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 319.504 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 322.781 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 326.058 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 327.696 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 330.973 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 334.25 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 337.527 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 340.804 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 342.443 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 345.72 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 348.997 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 352.274 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 355.551 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 358.828 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 360.466 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 363.743 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 367.02 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 370.297 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 373.574 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 376.851 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 378.489 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 381.766 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 385.043 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 388.32 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 391.597 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 394.874 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 396.513 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 399.79 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 403.067 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 406.344 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 409.62 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 412.897 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 414.536 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 417.813 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 421.09 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 424.367 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 427.644 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 430.921 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 432.559 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 435.836 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 439.113 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 442.39 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 445.667 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 448.944 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 450.583 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 453.859 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 457.136 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 460.413 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 463.69 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 466.967 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 468.606 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 471.883 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 475.16 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 478.437 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 481.714 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 484.991 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 486.629 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 489.906 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 493.183 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 496.46 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 499.737 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 503.014 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 504.652 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 507.929 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 511.206 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 514.483 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 517.76 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 521.037 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 522.676 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 525.953 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 529.23 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 532.507 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 535.784 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 539.061 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 540.699 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 543.976 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 547.253 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 550.53 208.604 :M f0_12 sf (B)S gR gS .61 .614 scale 558.722 211.864 :M f0_7 sf (0)S gR gS .61 .614 scale 561.999 208.604 :M f0_12 sf (:)S gR gS .61 .614 scale 565.276 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 568.553 208.604 :M f0_12 sf (X)S gR gS .61 .614 scale 576.746 208.604 :M f0_12 sf (,)S gR gS .61 .614 scale 581.661 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 584.938 208.604 :M f0_12 sf (O)S gR gS .61 .614 scale 591.492 211.864 :M f0_7 sf (1)S gR gS .61 .614 scale 596.407 208.604 :M f0_12 sf (,)S gR gS .61 .614 scale 599.684 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 602.961 208.604 :M f0_12 sf (O)S gR gS .61 .614 scale 611.154 211.864 :M f0_7 sf (2)S gR gS .61 .614 scale 616.069 208.604 :M f0_12 sf (,)S gR gS .61 .614 scale 619.346 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 622.623 208.604 :M f0_12 sf (O)S gR gS .61 .614 scale 630.816 211.864 :M f0_7 sf (3)S gR gS .61 .614 scale 634.092 208.604 :M f0_12 sf (,)S gR gS .61 .614 scale 639.008 208.604 :M f0_12 sf ( )S gR gS .61 .614 scale 640.646 208.604 :M f0_12 sf (Y)S gR gS .61 .614 scale 237.58 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 240.857 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 244.134 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 247.411 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 265.434 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 268.711 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 273.626 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 276.903 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 280.18 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 283.457 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 286.734 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 290.011 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 291.65 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 294.927 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 298.204 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 301.481 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 304.758 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 308.035 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 309.673 220.012 :M f0_12 sf (S)S gR gS .61 .614 scale 317.865 223.272 :M f0_7 sf (1)S gR gS .61 .614 scale 321.142 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 324.419 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 327.696 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 330.973 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 332.612 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 335.889 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 339.166 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 342.443 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 345.72 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 348.997 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 350.635 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 353.912 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 357.189 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 360.466 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 363.743 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 367.02 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 368.658 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 371.935 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 375.212 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 378.489 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 381.766 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 385.043 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 386.682 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 389.959 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 393.236 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 396.513 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 399.79 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 403.067 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 404.705 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 407.982 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 411.259 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 414.536 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 417.813 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 419.451 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 422.728 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 426.005 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 429.282 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 432.559 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 435.836 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 437.475 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 440.752 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 444.029 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 447.306 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 450.583 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 453.859 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 455.498 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 458.775 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 462.052 220.012 :M f0_12 sf ( )S gR gS .61 .614 scale 465.329 220.012 :M f0_12 sf (O)S gR gS .61 .614 scale 473.521 223.272 :M f0_7 sf (3)S gR gS .61 .614 scale 237.58 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 240.857 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 244.134 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 247.411 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 265.434 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 268.711 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 273.626 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 276.903 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 280.18 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 283.457 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 286.734 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 290.011 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 291.65 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 294.927 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 298.204 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 301.481 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 304.758 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 308.035 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 309.673 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 312.95 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 316.227 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 319.504 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 322.781 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 326.058 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 327.696 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 330.973 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 334.25 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 337.527 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 340.804 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 342.443 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 345.72 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 348.997 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 352.274 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 355.551 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 358.828 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 360.466 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 363.743 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 367.02 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 370.297 231.42 :M f4_12 sf (G)S gR gS .61 .614 scale 378.489 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 381.766 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 385.043 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 388.32 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 391.597 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 394.874 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 396.513 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 399.79 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 403.067 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 406.344 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 409.62 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 412.897 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 414.536 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 417.813 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 421.09 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 424.367 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 427.644 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 430.921 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 432.559 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 435.836 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 439.113 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 442.39 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 445.667 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 448.944 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 450.583 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 453.859 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 457.136 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 460.413 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 463.69 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 466.967 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 468.606 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 471.883 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 475.16 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 478.437 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 481.714 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 484.991 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 486.629 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 489.906 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 493.183 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 496.46 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 499.737 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 503.014 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 504.652 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 507.929 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 511.206 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 514.483 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 517.76 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 521.037 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 522.676 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 525.953 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 529.23 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 532.507 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 535.784 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 539.061 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 540.699 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 543.976 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 547.253 231.42 :M f4_12 sf ( )S gR gS .61 .614 scale 550.53 231.42 :M f0_12 sf (B)S gR gS .61 .614 scale 558.722 234.68 :M f0_7 sf (1)S gR gS .61 .614 scale 561.999 231.42 :M f0_12 sf (:)S gR gS .61 .614 scale 565.276 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 568.553 231.42 :M f0_12 sf (X)S gR gS .61 .614 scale 576.746 231.42 :M f0_12 sf (,)S gR gS .61 .614 scale 581.661 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 584.938 231.42 :M f0_12 sf (O)S gR gS .61 .614 scale 591.492 234.68 :M f0_7 sf (1)S gR gS .61 .614 scale 596.407 231.42 :M f0_12 sf (,)S gR gS .61 .614 scale 599.684 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 602.961 231.42 :M f0_12 sf (O)S gR gS .61 .614 scale 611.154 234.68 :M f0_7 sf (3)S gR gS .61 .614 scale 616.069 231.42 :M f0_12 sf (,)S gR gS .61 .614 scale 619.346 231.42 :M f0_12 sf ( )S gR gS .61 .614 scale 622.623 231.42 :M f0_12 sf (Y)S gR gS .61 .614 scale 237.58 255.866 :M f0_12 sf (X)S gR gS .61 .614 scale 247.411 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 255.866 :M f0_12 sf (o)S gR gS .61 .614 scale 268.711 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 255.866 :M f0_12 sf (O)S gR gS .61 .614 scale 278.542 259.126 :M f0_7 sf (1)S gR gS .61 .614 scale 283.457 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 286.734 255.866 :M f0_12 sf (o)S gR gS .61 .614 scale 291.65 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 294.927 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 298.204 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 301.481 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 304.758 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 308.035 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 309.673 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 312.95 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 316.227 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 319.504 255.866 :M f0_12 sf (O)S gR gS .61 .614 scale 327.696 259.126 :M f0_7 sf (2)S gR gS .61 .614 scale 330.973 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 334.25 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 337.527 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 340.804 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 342.443 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 345.72 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 348.997 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 352.274 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 355.551 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 358.828 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 360.466 255.866 :M f0_12 sf (O)S gR gS .61 .614 scale 368.658 259.126 :M f0_7 sf (3)S gR gS .61 .614 scale 373.574 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 376.851 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 378.489 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 381.766 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 385.043 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 388.32 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 391.597 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 394.874 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 396.513 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 399.79 255.866 :M f0_12 sf ( )S gR gS .61 .614 scale 403.067 255.866 :M f0_12 sf (Y)S gR gS .61 .614 scale 237.58 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 240.857 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 244.134 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 247.411 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 265.434 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 268.711 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 267.274 :M f0_12 sf ( )S gR gS .61 .614 scale 273.626 267.274 :M f0_12 sf (o)S gR gS .61 .614 scale 237.58 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 240.857 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 244.134 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 247.411 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 250.688 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 253.965 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 255.603 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 258.88 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 262.157 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 265.434 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 268.711 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 271.988 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 273.626 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 276.903 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 280.18 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 283.457 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 286.734 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 290.011 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 291.65 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 294.927 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 298.204 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 301.481 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 304.758 291.72 :M f0_12 sf ( )S gR gS .61 .614 scale 308.035 291.72 :M f0_12 sf (P)S gR gS .61 .614 scale 314.589 291.72 :M f0_12 sf (A)S gR gS .61 .614 scale 322.781 291.72 :M f0_12 sf (G)S gR gS .61 .614 scale 330.973 291.72 :M f0_12 sf <28>S gR gS .61 .614 scale 335.889 291.72 :M f4_12 sf (G)S gR gS .61 .614 scale 344.081 291.72 :M f0_12 sf <29>S gR gR gS 145 107 260 81 rC 154 111.75 -.75 .75 161.75 111 .75 154 111 @a np 155 112 :M 155 109 :L 152 111 :L 155 112 :L .75 lw eofill -.75 -.75 155.75 112.75 .75 .75 155 109 @b -.75 -.75 152.75 111.75 .75 .75 155 109 @b 152 111.75 -.75 .75 155.75 112 .75 152 111 @a 176 111.75 -.75 .75 184.75 111 .75 176 111 @a np 184 110 :M 184 113 :L 186 111 :L 184 110 :L eofill -.75 -.75 184.75 113.75 .75 .75 184 110 @b -.75 -.75 184.75 113.75 .75 .75 186 111 @b 184 110.75 -.75 .75 186.75 111 .75 184 110 @a 202 112.75 -.75 .75 210.75 112 .75 202 112 @a np 203 113 :M 203 111 :L 201 112 :L 203 113 :L eofill -.75 -.75 203.75 113.75 .75 .75 203 111 @b -.75 -.75 201.75 112.75 .75 .75 203 111 @b 201 112.75 -.75 .75 203.75 113 .75 201 112 @a 251 111.75 -.75 .75 258.75 111 .75 251 111 @a np 251 113 :M 251 110 :L 249 111 :L 251 113 :L eofill -.75 -.75 251.75 113.75 .75 .75 251 110 @b -.75 -.75 249.75 111.75 .75 .75 251 110 @b 249 111.75 -.75 .75 251.75 113 .75 249 111 @a 298 111.75 -.75 .75 305.75 111 .75 298 111 @a np 298 113 :M 298 110 :L 296 111 :L 298 113 :L eofill -.75 -.75 298.75 113.75 .75 .75 298 110 @b -.75 -.75 296.75 111.75 .75 .75 298 110 @b 296 111.75 -.75 .75 298.75 113 .75 296 111 @a 224 112.75 -.75 .75 232.75 112 .75 224 112 @a np 232 111 :M 232 113 :L 233 112 :L 232 111 :L eofill -.75 -.75 232.75 113.75 .75 .75 232 111 @b -.75 -.75 232.75 113.75 .75 .75 233 112 @b 232 111.75 -.75 .75 233.75 112 .75 232 111 @a 271 112.75 -.75 .75 279.75 112 .75 271 112 @a np 279 111 :M 279 113 :L 280 112 :L 279 111 :L eofill -.75 -.75 279.75 113.75 .75 .75 279 111 @b -.75 -.75 279.75 113.75 .75 .75 280 112 @b 279 111.75 -.75 .75 280.75 112 .75 279 111 @a -.75 -.75 193.75 127.75 .75 .75 193 118 @b np 194 127 :M 191 127 :L 193 128 :L 194 127 :L eofill 191 127.75 -.75 .75 194.75 127 .75 191 127 @a 191 127.75 -.75 .75 193.75 128 .75 191 127 @a -.75 -.75 193.75 128.75 .75 .75 194 127 @b -.75 -.75 289.75 125.75 .75 .75 289 117 @b np 290 125 :M 288 125 :L 289 127 :L 290 125 :L eofill 288 125.75 -.75 .75 290.75 125 .75 288 125 @a 288 125.75 -.75 .75 289.75 127 .75 288 125 @a -.75 -.75 289.75 127.75 .75 .75 290 125 @b 240 117.75 -.75 .75 282.75 131 .75 240 117 @a np 281 130 :M 280 132 :L 283 132 :L 281 130 :L eofill -.75 -.75 280.75 132.75 .75 .75 281 130 @b 280 132.75 -.75 .75 283.75 132 .75 280 132 @a 281 130.75 -.75 .75 283.75 132 .75 281 130 @a 153 155.75 -.75 .75 160.75 155 .75 153 155 @a np 154 156 :M 154 154 :L 151 155 :L 154 156 :L eofill -.75 -.75 154.75 156.75 .75 .75 154 154 @b -.75 -.75 151.75 155.75 .75 .75 154 154 @b 151 155.75 -.75 .75 154.75 156 .75 151 155 @a 177 155.75 -.75 .75 188.75 155 .75 177 155 @a np 188 154 :M 188 156 :L 190 155 :L 188 154 :L eofill -.75 -.75 188.75 156.75 .75 .75 188 154 @b -.75 -.75 188.75 156.75 .75 .75 190 155 @b 188 154.75 -.75 .75 190.75 155 .75 188 154 @a 207 155.75 -.75 .75 215.75 155 .75 207 155 @a np 215 154 :M 215 157 :L 216 155 :L 215 154 :L eofill -.75 -.75 215.75 157.75 .75 .75 215 154 @b -.75 -.75 215.75 157.75 .75 .75 216 155 @b 215 154.75 -.75 .75 216.75 155 .75 215 154 @a 234 156.75 -.75 .75 241.75 156 .75 234 156 @a np 235 157 :M 235 155 :L 232 156 :L 235 157 :L eofill -.75 -.75 235.75 157.75 .75 .75 235 155 @b -.75 -.75 232.75 156.75 .75 .75 235 155 @b 232 156.75 -.75 .75 235.75 157 .75 232 156 @a 90 180 51 15 195 165 @n -.75 -.75 223.75 164.75 .75 .75 223 161 @b np 222 162 :M 224 162 :L 223 159 :L 222 162 :L eofill 222 162.75 -.75 .75 224.75 162 .75 222 162 @a 223 159.75 -.75 .75 224.75 162 .75 223 159 @a -.75 -.75 222.75 162.75 .75 .75 223 159 @b 0 90 58 16 193.5 164.5 @n gR gS 0 0 552 730 rC 254 201 :M f2_10 sf 3.634 .363(Figure 3)J 59 216 :M f0_10 sf .208 .021(First form the following sequence of vertices. A\(0\) = X. If A\(n)J f1_10 sf .078(-)A f0_10 sf .119 .012(1\) )J f1_10 sf .078A f0_10 sf .162 .016(\312Y, )J 352 216 :M .202 .02(then )J 373 216 :M -.094(A\(n\) )A 395 216 :M .694 .069(is )J 406 216 :M .218 .022(the )J 422 216 :M .266 .027(first )J 442 216 :M -.081(vertex )A 470 216 :M .417 .042(on )J 484 216 :M f4_10 sf (U)S 59 228 :M f0_10 sf -.162(after )A 80 228 :M .022(A\(n)A f1_10 sf (-)S f0_10 sf (1\) )S 113 228 :M .123 .012(such )J 135 228 :M .361 .036(that )J 154 228 :M -.044(either )A 180 228 :M .786 .079(it )J 190 228 :M .694 .069(is )J 201 228 :M .056 .006(a )J 209 228 :M -.114(non-collider )A 260 228 :M .417 .042(on )J 274 228 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 285 228 :M -.313(and )A 302 228 :M .694 .069(is )J 313 228 :M .601 .06(in )J 325 228 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 340 228 :M .144 .014(or )J 353 228 :M .786 .079(it )J 364 228 :M .694 .069(is )J 376 228 :M .056 .006(a )J 385 228 :M -.13(collider )A 419 228 :M .417 .042(on )J 434 228 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 446 228 :M -.313(and )A 464 228 :M .555 .056(not )J 482 228 :M -.439(an)A 59 240 :M .116 .012(ancestor of )J f2_10 sf (S)S f0_10 sf .092 .009(. The last vertex in the sequence is Y because Y is in )J f2_10 sf .062(O)A f0_10 sf .065 .006( and )J 355 240 :M .555 .056(not )J 372 240 :M .056 .006(a )J 380 240 :M -.13(collider )A 413 240 :M .417 .042(on )J 427 240 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 441 240 :M .361 .036(Suppose )J 479 240 :M -.108(the)A 59 252 :M .129 .013(length of the sequence is n, i.e. A\(n\) = Y.)J 59 267 :M .207 .021(Note that for 1 )J cF f1_10 sf .021A sf .207 .021( i )J cF f1_10 sf .021A sf .207 .021( n, if A\(i\) is a collider on )J f4_10 sf .156(U)A f0_10 sf .219 .022( then it is an ancestor of )J f2_10 sf .144(Z)A f0_10 sf .292 .029(, because )J 400 267 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 411 267 :M -.181(d-connects )A 456 267 :M .255 .026(X )J 467 267 :M -.313(and )A 484 267 :M (Y)S 59 279 :M 1.149 .115(given )J f2_10 sf .496(Z)A f0_10 sf .169 .017( )J f1_10 sf .571A f0_10 sf .169 .017( )J f2_10 sf .413(S)A f0_10 sf (.)S 59 294 :M .361 .036(Suppose )J 98 294 :M -.038(A\(i\) )A 119 294 :M .694 .069(is )J 132 294 :M .601 .06(in )J 146 294 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 163 294 :M .555 .056(but )J 182 294 :M .555 .056(not )J 201 294 :M .056 .006(a )J 211 294 :M -.13(collider )A 246 294 :M .417 .042(on )J 262 294 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 278 294 :M (Then )S 304 294 :M -.052(for )A 321 294 :M .454 .045(1 )J 332 294 :M .782 .078(< )J 344 294 :M .656 .066(i )J 353 294 :M cF f1_10 sf .092A sf .92 .092( )J 365 294 :M .369(n)A f1_10 sf .405(-)A f0_10 sf .615 .061(1, )J 390 294 :M -.044(either )A 418 294 :M f4_10 sf -.077(U)A f0_10 sf -.044(\(A\(i)A f1_10 sf -.059(-)A f0_10 sf -.044(1\),A\(i\)\) )A 483 294 :M -.328(or)A 59 306 :M f4_10 sf .091(U)A f0_10 sf .195 .02(\(A\(i\),A\(i+1\)\) is out A\(i\). Suppose without loss of generality that )J f4_10 sf .091(U)A f0_10 sf .052(\(A\(i)A f1_10 sf .069(-)A f0_10 sf .16 .016(1\),A\(i\)\) is out of A\(i\). Then )J 475 306 :M -.218(A\(i\))A 59 318 :M .319 .032(is an ancestor of )J f2_10 sf .152(S)A f0_10 sf .158 .016( or )J 149 318 :M .054(A\(i)A f1_10 sf .066(-)A f0_10 sf .109 .011(1\) )J 180 318 :M -.163(because )A 214 318 :M -.044(either )A 240 318 :M f4_10 sf -.077(U)A f0_10 sf -.044(\(A\(i)A f1_10 sf -.059(-)A f0_10 sf -.044(1\),A\(i\)\) )A 303 318 :M .098 .01(contains )J 340 318 :M .417 .042(no )J 354 318 :M -.103(colliders )A 391 318 :M .601 .06(in )J 403 318 :M .043 .004(which )J 431 318 :M -.176(case )A 451 318 :M -.038(A\(i\) )A 471 318 :M .694 .069(is )J 482 318 :M -.439(an)A 59 330 :M (ancestor of A\(i)S f1_10 sf (-)S f0_10 sf .009 .001(1\), or it does contain a collider, in which case )J 310 330 :M .218 .022(the )J 326 330 :M .266 .027(first )J 346 330 :M -.13(collider )A 379 330 :M .694 .069(is )J 390 330 :M .051 .005(an )J 403 330 :M -.101(ancestor )A 439 330 :M .144 .014(of )J 451 330 :M .056 .006(a )J 459 330 :M -.152(member)A 59 342 :M .099 .01(of )J f2_10 sf .066(S)A f0_10 sf .145 .014(, and A\(i)J f1_10 sf .065(-)A f0_10 sf .152 .015(1\) is an ancestor of the first collider. Similarly, if )J f4_10 sf .086(U)A f0_10 sf .288 .029(\(A\(i\),A\(i+1\)\) )J 382 342 :M .694 .069(is )J 393 342 :M .555 .056(out )J 410 342 :M -.038(A\(i\) )A 430 342 :M .202 .02(then )J 451 342 :M -.038(A\(i\) )A 471 342 :M .694 .069(is )J 482 342 :M -.439(an)A 59 354 :M -.024(ancestor of A\(i+1\) )A 136 354 :M .144 .014(or )J 148 354 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(. )J 162 354 :M .784 .078(So )J 177 354 :M .328 .033(if )J 187 354 :M -.038(A\(i\) )A 207 354 :M .694 .069(is )J 218 354 :M .601 .06(in )J 230 354 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 245 354 :M .555 .056(but )J 262 354 :M .555 .056(not )J 279 354 :M .056 .006(a )J 287 354 :M -.13(collider )A 320 354 :M .417 .042(on )J 334 354 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 348 354 :M .202 .02(then )J 369 354 :M -.038(A\(i\) )A 389 354 :M .694 .069(is )J 400 354 :M .051 .005(an )J 413 354 :M -.101(ancestor )A 449 354 :M .144 .014(of )J 461 354 :M .054(A\(i)A f1_10 sf .066(-)A f0_10 sf .066(1\),)A 59 366 :M .645 .065(A\(i+1\) or )J f2_10 sf .265(S)A f0_10 sf (.)S 59 381 :M .013 .001(Now form the sequence of vertices where for 1 )J cF f1_10 sf .001A sf .013 .001( i )J cF f1_10 sf .001A sf .013 .001( n, B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf (\(i\) = )S 312 381 :M -.038(A\(i\) )A 332 381 :M .328 .033(if )J 342 381 :M -.038(A\(i\) )A 362 381 :M .601 .06(in )J 374 381 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 389 381 :M -.313(and )A 406 381 :M -.041(otherwise )A 448 381 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 470 381 :M -.14(= )A 479 381 :M .389(O)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf (,)S 59 393 :M -.185(where )A 86 393 :M .389(O)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 99 393 :M .694 .069(is )J 110 393 :M .218 .022(the )J 126 393 :M .266 .027(first )J 146 393 :M -.081(vertex )A 174 393 :M .601 .06(in )J 186 393 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 198 393 :M .417 .042(on )J 212 393 :M .056 .006(a )J 220 393 :M .222 .022(shortest )J 255 393 :M .202 .02(path )J 277 393 :M .047 .005(from )J 301 393 :M -.038(A\(i\) )A 322 393 :M .601 .06(to )J 335 393 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(. )J 350 393 :M .117 .012(\(Such )J 378 393 :M .056 .006(a )J 387 393 :M .202 .02(path )J 409 393 :M .454 .045(exists )J 437 393 :M -.163(because )A 472 393 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 484 393 :M -1.328(d-)A 59 405 :M .247 .025(connects X and Y given )J f2_10 sf .122(Z)A f0_10 sf ( )S f1_10 sf .141A f0_10 sf ( )S f2_10 sf .102(S)A f0_10 sf .193 .019(, and no A\(i\) that is a collider on )J f4_10 sf .132(U)A f0_10 sf .186 .019( is an ancestor of )J f2_10 sf .102(S)A f0_10 sf .196 .02(.\) We will )J 449 405 :M .216 .022(now )J 470 405 :M -.036(show)A 59 417 :M .115 .011(that for 1 )J cF f1_10 sf .011A sf .115 .011( i )J cF f1_10 sf .011A sf .115 .011( n, there is an inducing path between B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .106 .011(\(i\) and )J 308 417 :M (B)S f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.026(\(i+1\).The )A 359 417 :M .202 .02(path )J 380 417 :M -.116(between )A 416 417 :M -.038(A\(i\) )A 436 417 :M -.313(and )A 453 417 :M .13 .013(A\(i+1\) )J 484 417 :M -1.328(d-)A 59 429 :M .33 .033(connects A\(i\) and A\(i+1\) given \()J f2_10 sf .096(Ancestors)A f0_10 sf .101(\({B)A f0_6 sf 0 2 rm .062(0)A 0 -2 rm f0_10 sf .077(\(i\),B)A f0_6 sf 0 2 rm .062(0)A 0 -2 rm f0_10 sf .331 .033(\(i+1\)} )J f1_10 sf .158A f0_10 sf .047 .005( )J f2_10 sf .114(S)A f0_10 sf .1 .01(\) )J f1_10 sf .158A f0_10 sf .047 .005( )J f2_10 sf .16(O)A f0_10 sf .1 .01(\) )J f1_10 sf .158A f0_10 sf .051 .005( )J 362 429 :M f2_10 sf (S)S f0_10 sf .103 .01(\)\\{A\(i\),A\(i+1\)}, )J 436 429 :M -.163(because )A 470 429 :M -.301(every)A 59 441 :M -.13(collider )A 92 441 :M .417 .042(on )J 106 441 :M f4_10 sf -.094(U)A f0_10 sf -.056(\(A\(i\),A\(i+1\)\) )A 169 441 :M .694 .069(is )J 180 441 :M .051 .005(an )J 193 441 :M -.101(ancestor )A 229 441 :M .144 .014(of )J 241 441 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(, )J 255 441 :M -.313(and )A 272 441 :M .417 .042(no )J 286 441 :M -.114(non-collider )A 337 441 :M .417 .042(on )J 351 441 :M f4_10 sf -.094(U)A f0_10 sf -.056(\(A\(i\),A\(i+1\)\) )A 414 441 :M .694 .069(is )J 425 441 :M .601 .06(in )J 437 441 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 449 441 :M f1_10 sf .62A f0_10 sf .202 .02( )J 461 441 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(. )J 476 441 :M -.273(The)A 59 453 :M -.022(path from A\(i\) to O)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf -.021( \(if there is one\) d-connects A\(i\) and )A 287 453 :M .389(O)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 300 453 :M .189 .019(given )J 326 453 :M .08<28>A f2_10 sf .113(Ancestors)A f0_10 sf .119(\({B)A f0_6 sf 0 2 rm .072(0)A 0 -2 rm f0_10 sf .089(\(i\),B)A f0_6 sf 0 2 rm .072(0)A 0 -2 rm f0_10 sf .411 .041(\(i+1\)} )J 442 453 :M f1_10 sf .62A f0_10 sf .202 .02( )J 454 453 :M f2_10 sf .788(S)A f0_10 sf .751 .075(\) )J 468 453 :M f1_10 sf .62A f0_10 sf .202 .02( )J 480 453 :M f2_10 sf -.074(O)A f0_10 sf <29>S 59 465 :M f1_10 sf S f0_10 sf ( )S f2_10 sf (S)S f0_10 sf .014(\)\\{A\(i\),O)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .044 .004(} because by )J 169 465 :M .032 .003(construction )J 222 465 :M .786 .079(it )J 232 465 :M .694 .069(is )J 243 465 :M .056 .006(a )J 251 465 :M -.337(directed )A 284 465 :M .202 .02(path )J 305 465 :M .361 .036(that )J 324 465 :M .098 .01(contains )J 361 465 :M .417 .042(no )J 375 465 :M -.044(member )A 411 465 :M .144 .014(of )J 423 465 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 435 465 :M -.099(except )A 464 465 :M -.052(for )A 479 465 :M .389(O)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf (,)S 59 477 :M .294 .029(and no member of )J f2_10 sf .124(S)A f0_10 sf .101 .01(. )J 148 477 :M .891 .089(Similarly, )J 193 477 :M .218 .022(the )J 209 477 :M .202 .02(path )J 230 477 :M .047 .005(from )J 253 477 :M .13 .013(A\(i+1\) )J 284 477 :M .601 .06(to )J 296 477 :M .093(O)A f0_6 sf 0 2 rm .035(i+1)A 0 -2 rm f0_10 sf ( )S 315 477 :M .051 .005(\(if )J 328 477 :M -.097(there )A 351 477 :M .694 .069(is )J 362 477 :M -.067(one\) )A 383 477 :M -.181(d-connects )A 428 477 :M .13 .013(A\(i+1\) )J 459 477 :M -.313(and )A 476 477 :M -.128(O)A f0_6 sf 0 2 rm -.071(i+1)A 0 -2 rm 59 489 :M f0_10 sf .189 .019(given )J 85 489 :M .08<28>A f2_10 sf .113(Ancestors)A f0_10 sf .119(\({B)A f0_6 sf 0 2 rm .072(0)A 0 -2 rm f0_10 sf .089(\(i\),B)A f0_6 sf 0 2 rm .072(0)A 0 -2 rm f0_10 sf .411 .041(\(i+1\)} )J 201 489 :M f1_10 sf .62A f0_10 sf .202 .02( )J 213 489 :M f2_10 sf .788(S)A f0_10 sf .751 .075(\) )J 227 489 :M f1_10 sf .62A f0_10 sf .202 .02( )J 239 489 :M f2_10 sf .225(O)A f0_10 sf .153 .015(\) )J 254 489 :M f1_10 sf .62A f0_10 sf .202 .02( )J 266 489 :M f2_10 sf .052(S)A f0_10 sf .041(\)\\{A\(i+1\),O)A f0_6 sf 0 2 rm .025(i+1)A 0 -2 rm f0_10 sf .077 .008(}. )J 339 489 :M -.085(By )A 354 489 :M (Lemma )S 388 489 :M .454 .045(1 )J 397 489 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 419 489 :M -.313(and )A 436 489 :M -.05(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 469 489 :M -.235(are )A 484 489 :M -1.328(d-)A 59 501 :M .417 .042(connected given )J f2_10 sf .104(Ancestors)A f0_10 sf .109(\({B)A f0_6 sf 0 2 rm .067(0)A 0 -2 rm f0_10 sf .083(\(i\),B)A f0_6 sf 0 2 rm .067(0)A 0 -2 rm f0_10 sf .357 .036(\(i+1\)} )J f1_10 sf .17A f0_10 sf .05 .005( )J f2_10 sf .123(S)A f0_10 sf .108 .011(\) )J f1_10 sf .17A f0_10 sf .05 .005( )J f2_10 sf .173(O)A f0_10 sf .117 .012(\) )J 287 501 :M f1_10 sf .62A f0_10 sf .202 .02( )J 299 501 :M f2_10 sf .129(S)A f0_10 sf .419 .042(\)\\{A\(i\), )J 339 501 :M .138 .014(A\(i+1}, )J 374 501 :M -.313(and )A 391 501 :M .417 .042(by )J 405 501 :M (Lemma )S 439 501 :M .454 .045(5 )J 448 501 :M -.097(there )A 471 501 :M .694 .069(is )J 482 501 :M -.439(an)A 59 513 :M -.085(inducing path between B)A f0_6 sf 0 2 rm -.059(0)A 0 -2 rm f0_10 sf -.08(\(i\) and )A 189 513 :M (B)S f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .116 .012(\(i+1\). )J 225 513 :M -.259(Because )A 260 513 :M -.052(for )A 275 513 :M .454 .045(1 )J 284 513 :M cF f1_10 sf .092A sf .92 .092( )J 294 513 :M .656 .066(i )J 301 513 :M cF f1_10 sf .092A sf .92 .092( )J 311 513 :M .281(n)A f1_10 sf .309(-)A f0_10 sf .383 .038(1 )J 331 513 :M -.097(there )A 354 513 :M .694 .069(is )J 365 513 :M .051 .005(an )J 378 513 :M -.062(inducing )A 416 513 :M .202 .02(path )J 437 513 :M -.116(between )A 473 513 :M -.385(B)A f0_6 sf 0 2 rm -.173(0)A 0 -2 rm f0_10 sf -.272(\(i\))A 59 525 :M .186 .019(and B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .143 .014(\(i+1\), there is a path B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .166 .017( in MAG\()J f4_10 sf .085(G)A f0_10 sf <28>S f2_10 sf .092(O)A f0_10 sf (,)S f2_10 sf .066(S)A f0_10 sf (,)S f2_10 sf .079(L)A f0_10 sf .152 .015(\)\) on which B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .108 .011(\(i\) is the i)J f0_6 sf 0 -4 rm .028(th)A 0 4 rm f0_10 sf .208 .021( vertex.)J 59 540 :M (If B)S f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .029 .003(\(i\) is not a non)J f1_10 sf (-)S f0_10 sf .052 .005(collider )J 176 540 :M .417 .042(on )J 190 540 :M f4_10 sf .461(U)A f0_10 sf .29 .029(, )J 204 540 :M .047 .005(any )J 222 540 :M -.344(edge )A 243 540 :M .361 .036(that )J 262 540 :M .098 .01(contains )J 299 540 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 321 540 :M .417 .042(on )J 335 540 :M .047 .005(any )J 353 540 :M .144 .014(of )J 365 540 :M .218 .022(the )J 381 540 :M .263 .026(paths )J 406 540 :M -.207(used )A 427 540 :M .601 .06(to )J 439 540 :M -.017(construct )A 479 540 :M -.108(the)A 59 552 :M -.023(inducing path between B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.021(\(i\) and B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.021(\(i+1\) or B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf -.019(1\), is into B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.022(\(i\). Hence the inducing )A 399 552 :M .202 .02(path )J 420 552 :M -.116(between )A 456 552 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 478 552 :M -.719(and)A 59 564 :M -.093(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.058(\(i+1\) and the inducing path between B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.055(\(i\) and B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.042(\(i)A f1_10 sf -.076(-)A f0_10 sf -.054(1\) are into B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.055(\(i\).)A 59 579 :M .353 .035(If B)J f0_6 sf 0 2 rm .094(0)A 0 -2 rm f0_10 sf .262 .026(\(i\) is on )J f4_10 sf .225(U)A f0_10 sf .322 .032( but not a collider on )J f4_10 sf .225(U)A f0_10 sf .35 .035(, then B)J f0_6 sf 0 2 rm .094(0)A 0 -2 rm f0_10 sf .27 .027(\(i\) is not in )J f2_10 sf .208(Z)A f0_10 sf .071 .007( )J f1_10 sf .24A f0_10 sf .071 .007( )J f2_10 sf .174(S)A f0_10 sf .078 .008( )J 329 579 :M -.163(because )A 363 579 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 374 579 :M -.181(d-connects )A 419 579 :M .255 .026(X )J 430 579 :M -.313(and )A 447 579 :M .255 .026(Y )J 458 579 :M .189 .019(given )J 484 579 :M f2_10 sf (Z)S 59 591 :M f1_10 sf .62A f0_10 sf .202 .02( )J 71 591 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(. )J 85 591 :M .144 .014(In )J 97 591 :M -.086(addition, )A 135 591 :M -.044(either )A 161 591 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 183 591 :M .694 .069(is )J 194 591 :M .051 .005(an )J 208 591 :M -.101(ancestor )A 245 591 :M .144 .014(of )J 258 591 :M .079(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .036(\(i)A f1_10 sf .065(-)A f0_10 sf .121 .012(1\), )J 295 591 :M -.05(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 329 591 :M .144 .014(or )J 342 591 :M f2_10 sf 1.285(S)A f0_10 sf 1.051 .105(, )J 357 591 :M -.163(because )A 392 591 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 415 591 :M -.14(= )A 425 591 :M .231 .023(A\(i\), )J 449 591 :M -.038(A\(i\) )A 470 591 :M .694 .069(is )J 482 591 :M -.439(an)A 59 603 :M .165 .017(ancestor of A\(i)J f1_10 sf .053(-)A f0_10 sf .118 .012(1\), A\(i+1\) or )J f2_10 sf .054(S)A f0_10 sf .117 .012(, and A\(i)J f1_10 sf .053(-)A f0_10 sf .114 .011(1\) is an ancestor of B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .029(\(i)A f1_10 sf .053(-)A f0_10 sf .117 .012(1\), and A\(i+1\) is )J 398 603 :M .051 .005(an )J 411 603 :M -.101(ancestor )A 447 603 :M .144 .014(of )J 459 603 :M -.05(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.034(\(i+1\).)A 59 615 :M .395 .04(It follows that B)J f0_6 sf 0 2 rm .085(0)A 0 -2 rm f0_10 sf .316 .032(\(i\) is not a collider on B)J f0_6 sf 0 2 rm .085(0)A 0 -2 rm f0_10 sf .244 .024(, and is not in )J f2_10 sf .19(Z)A f0_10 sf .065 .006( )J f1_10 sf .219A f0_10 sf .065 .006( )J f2_10 sf .158(S)A f0_10 sf (.)S 59 630 :M .178 .018(By construction, if B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .13 .013(\(i\) is not a non-collider on )J f4_10 sf .078(U)A f0_10 sf .097 .01(, B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .087 .009(\(i\) is in )J 310 630 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 325 630 :M .051 .005(an )J 338 630 :M -.101(ancestor )A 374 630 :M .144 .014(of )J 386 630 :M f2_10 sf .76(Z)A f0_10 sf .518 .052(, )J 400 630 :M -.313(and )A 417 630 :M .555 .056(not )J 434 630 :M .051 .005(an )J 447 630 :M -.101(ancestor )A 483 630 :M -.328(of)A 59 642 :M f2_10 sf .081(S)A f0_10 sf .243 .024(. However, B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .159 .016(\(i\) may be )J 167 642 :M .601 .06(in )J 179 642 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 190 642 :M .555 .056(but )J 207 642 :M .555 .056(not )J 224 642 :M .051 .005(be )J 237 642 :M .056 .006(a )J 245 642 :M -.13(collider )A 278 642 :M .417 .042(on )J 292 642 :M -.093(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf ( )S 305 642 :M .601 .06(in )J 317 642 :M .153(MAG\()A f4_10 sf .165(G)A f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .175 .017(\)\) )J 391 642 :M .051 .005(\(if )J 404 642 :M .601 .06(in )J 416 642 :M f4_10 sf .457(G)A f0_10 sf .211<28>A f2_10 sf .492(O)A f0_10 sf .158(,)A f2_10 sf .352(S)A f0_10 sf .158(,)A f2_10 sf .422(L)A f0_10 sf .335 .034(\) )J 461 642 :M .786 .079(it )J 471 642 :M .694 .069(is )J 482 642 :M -.439(an)A 59 654 :M -.038(ancestor of either its predecessor or successor on )A f4_10 sf -.071(U)A f0_10 sf -.043(\). The )A 289 654 :M .325 .033(following )J 332 654 :M .327 .033(algorithm )J 375 654 :M -.053(removes )A 412 654 :M .388 .039(all )J 426 654 :M -.097(non-colliders )A 481 654 :M (on)S 59 666 :M .067(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .102 .01( which are in )J f2_10 sf .067(Z)A f0_10 sf (.)S 59 686 :M .233 .023(k = 0;)J endp %%Page: 10 10 %%BeginPageSetup initializepage (peter; page: 10 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 479 713 12 12 rC 479 722 :M f0_12 sf (10)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf -.153(Repeat)A 77 66 :M .1 .01(If there is a triple of )J 161 66 :M -.074(vertices )A 195 66 :M .079(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .036(\(i)A f1_10 sf .065(-)A f0_10 sf .121 .012(1\), )J 231 66 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.015(\(i\), )A 256 66 :M -.05(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 289 66 :M .123 .012(such )J 311 66 :M .361 .036(that )J 330 66 :M .218 .022(the )J 346 66 :M -.062(inducing )A 384 66 :M .263 .026(paths )J 409 66 :M -.116(between )A 445 66 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf (1\) )S 478 66 :M -.719(and)A 77 78 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.027(\(i\), and B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.028(\(i\) and B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.027(\(i+1\) collide at B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.026(\(i\), but B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.022(\(i\) is in )A f2_10 sf (Z)S f0_10 sf -.028( and an ancestor of B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf -.038(1\) )A 424 78 :M .144 .014(or )J 436 78 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .116 .012(\(i+1\), )J 472 78 :M -.145(form)A 77 90 :M (sequence B)S f0_6 sf 0 2 rm (k+1)S 0 -2 rm f0_10 sf ( by removing B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.001(\(i\) from the sequence \(i.e. for 1 )A cF f1_10 sf -.001A sf -.001( j < i, set B)A f0_6 sf 0 2 rm (k+1)S 0 -2 rm f0_10 sf (\(j\) = B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(j\), )S 429 90 :M -.313(and )A 446 90 :M -.052(for )A 461 90 :M .656 .066(i )J 468 90 :M cF f1_10 sf .092A sf .92 .092( )J 478 90 :M .656 .066(j )J 485 90 :M cF f1_10 sf S sf 77 102 :M (n)S f1_10 sf (-)S f0_10 sf (1, set B)S f0_6 sf 0 2 rm (k+1)S 0 -2 rm f0_10 sf (\(j\) = B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(j+1\)\);)S 77 117 :M .433 .043(k := k + 1;)J 59 132 :M .05 .005(until there is no such triple of vertices in the sequence B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (.)S 59 152 :M .139 .014(At each stage of the algorithm, if there is a )J 235 152 :M .249 .025(triple )J 260 152 :M .144 .014(of )J 272 152 :M -.074(vertices )A 306 152 :M .079(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .036(\(i)A f1_10 sf .065(-)A f0_10 sf .121 .012(1\), )J 342 152 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.015(\(i\), )A 367 152 :M -.05(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 400 152 :M .123 .012(such )J 422 152 :M .361 .036(that )J 441 152 :M .218 .022(the )J 457 152 :M -.142(inducing)A 59 164 :M -.075(paths between B)A f0_6 sf 0 2 rm -.051(k)A 0 -2 rm f0_10 sf -.052(\(i)A f1_10 sf -.093(-)A f0_10 sf -.073(1\) and B)A f0_6 sf 0 2 rm -.051(k)A 0 -2 rm f0_10 sf -.064(\(i\), and B)A f0_6 sf 0 2 rm -.051(k)A 0 -2 rm f0_10 sf -.067(\(i\) and B)A f0_6 sf 0 2 rm -.051(k)A 0 -2 rm f0_10 sf -.065(\(i+1\) collide at B)A f0_6 sf 0 2 rm -.051(k)A 0 -2 rm f0_10 sf -.061(\(i\), )A 339 164 :M .555 .056(but )J 356 164 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 378 164 :M .694 .069(is )J 389 164 :M .051 .005(an )J 402 164 :M -.101(ancestor )A 438 164 :M .144 .014(of )J 450 164 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf (1\) )S 483 164 :M -.328(or)A 59 176 :M .06(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .127 .013(\(i+1\), then by Lemma 1 through Lemma 5 there is an inducing path )J 345 176 :M -.116(between )A 381 176 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf (1\) )S 414 176 :M -.313(and )A 431 176 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .116 .012(\(i+1\). )J 467 176 :M -.384(Hence)A 59 188 :M -.002(for every k and each i, 1 )A cF f1_10 sf -.002A sf -.002( i )A cF f1_10 sf -.002A sf -.002( length of sequence B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.002(, there is an inducing path between B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i\) and B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i+1\). )S 485 188 :M -.106(It)A 59 200 :M .254 .025(follows that there is path B)J f0_6 sf 0 2 rm .055(k)A 0 -2 rm f0_10 sf .258 .026( in MAG\()J f4_10 sf .132(G)A f0_10 sf .061<28>A f2_10 sf .143(O)A f0_10 sf (,)S f2_10 sf .102(S)A f0_10 sf (,)S f2_10 sf .122(L)A f0_10 sf .205 .021(\)\) such that the i)J f0_6 sf 0 -4 rm .043(th)A 0 4 rm f0_10 sf .202 .02( vertex on the path is B)J f0_6 sf 0 2 rm .055(k)A 0 -2 rm f0_10 sf .073(\(i\).)A 59 215 :M .361 .036(Suppose )J 99 215 :M .266 .027(first )J 121 215 :M .361 .036(that )J 142 215 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 166 215 :M .694 .069(is )J 179 215 :M .056 .006(a )J 189 215 :M -.114(non-collider )A 242 215 :M .417 .042(on )J 258 215 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 275 215 :M -.188(We )A 295 215 :M .676 .068(will )J 318 215 :M .28 .028(show )J 346 215 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 371 215 :M .694 .069(is )J 385 215 :M .056 .006(a )J 396 215 :M -.114(non-collider )A 450 215 :M .417 .042(on )J 467 215 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf ( )S 483 215 :M .222(in)A 59 227 :M .11(MAG\()A f4_10 sf .119(G)A f0_10 sf .055<28>A f2_10 sf .129(O)A f0_10 sf (,)S f2_10 sf .092(S)A f0_10 sf (,)S f2_10 sf .11(L)A f0_10 sf .142 .014(\)\), and not in )J f2_10 sf .11(Z)A f0_10 sf ( )S f1_10 sf .127A f0_10 sf ( )S f2_10 sf .092(S)A f0_10 sf .206 .021( . We have already shown this for B)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .196 .02(. In addition, we )J 424 227 :M -.094(have )A 446 227 :M .261 .026(shown )J 476 227 :M (that)S 59 239 :M .328 .033(if )J 69 239 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 91 239 :M .694 .069(is )J 102 239 :M .056 .006(a )J 110 239 :M -.114(non-collider )A 161 239 :M .417 .042(on )J 175 239 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 186 239 :M -.044(either )A 212 239 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 234 239 :M .694 .069(is )J 245 239 :M .051 .005(an )J 258 239 :M -.101(ancestor )A 294 239 :M .144 .014(of )J 306 239 :M .079(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .036(\(i)A f1_10 sf .065(-)A f0_10 sf .121 .012(1\), )J 342 239 :M -.05(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 375 239 :M .144 .014(or )J 387 239 :M f2_10 sf 3.701 .37(S. )J 403 239 :M f0_10 sf .361 .036(Suppose )J 441 239 :M -.052(for )A 456 239 :M .454 .045(1 )J 466 239 :M cF f1_10 sf .092A sf .92 .092( )J 477 239 :M .656 .066(i )J 485 239 :M cF f1_10 sf S sf 59 251 :M .02(length\(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .04 .004(\), if B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .056 .006(\(i\) is a non-collider on )J f4_10 sf (U)S f0_10 sf .054 .005( either B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .054 .005(\(i\) is an ancestor of B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf .061 .006(1\), B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .053 .005(\(i+1\) or )J f2_10 sf (S.)S f0_10 sf .022 .002( It )J 471 251 :M .694 .069(is )J 482 251 :M -.439(an)A 59 263 :M -.004(ancestor of B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf (1\), B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i+1\) or )S f2_10 sf (S)S f0_10 sf (, unless B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.004(\(i\) is an ancestor of B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf (1\), and B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf -.005(1\) was removed at )A 479 263 :M -.108(the)A 59 275 :M (k)S f0_6 sf 0 -4 rm (th)S 0 4 rm f0_10 sf .114 .011( step of the algorithm, or B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .11 .011(\(i\) is an ancestor of B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .145 .015(\(i+1\), )J 310 275 :M -.313(and )A 327 275 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i+1\) )A 366 275 :M (was )S 385 275 :M -.212(removed )A 422 275 :M .236 .024(at )J 433 275 :M .218 .022(the )J 449 275 :M .342(k)A f0_6 sf 0 -4 rm .16(th)A 0 4 rm f0_10 sf .171 .017( )J 463 275 :M .281 .028(step )J 483 275 :M -.328(of)A 59 287 :M .253 .025(the algorithm. Suppose without loss of generality that B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm .051(-)A 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .131 .013(\(i\) is an )J 330 287 :M -.101(ancestor )A 366 287 :M .144 .014(of )J 378 287 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i+1\), )S 420 287 :M -.313(and )A 437 287 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i+1\) )A 476 287 :M -.273(was)A 59 299 :M .199 .02(removed at the k)J f0_6 sf 0 -4 rm .032(th)A 0 4 rm f0_10 sf .173 .017( step of the algorithm. It follows )J 266 299 :M .361 .036(that )J 285 299 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i+1\) )A 324 299 :M .694 .069(is )J 335 299 :M .051 .005(an )J 348 299 :M -.101(ancestor )A 384 299 :M .144 .014(of )J 396 299 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i+2\), )S 438 299 :M -.087(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.047(\(i\), )A 469 299 :M .144 .014(or )J 481 299 :M f2_10 sf 1.339(S)A f0_10 sf (.)S 59 311 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .057 .006(\(i+1\) is not an ancestor of B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .065 .007(\(i\) because )J f4_10 sf (G)S f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf (\) )S 286 311 :M .694 .069(is )J 297 311 :M -.039(acyclic. )A 331 311 :M -.089(Hence, )A 362 311 :M .786 .079(it )J 372 311 :M .298 .03(follows )J 406 311 :M .361 .036(that )J 425 311 :M .786 .079(it )J 435 311 :M .694 .069(is )J 446 311 :M .051 .005(an )J 459 311 :M -.187(ancestor)A 59 323 :M .168 .017(of either B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .14 .014(\(i+2\) or )J f2_10 sf .068(S)A f0_10 sf .145 .015(. It follows that B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .104 .01(\(i\) )J f1_10 sf .067A f0_10 sf .093 .009( B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .135 .013(\(i\) is an ancestor of )J f2_10 sf .068(S)A f0_10 sf .098 .01( or B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .172 .017(\(i+2\) )J f1_10 sf .067A f0_10 sf .093 .009( B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .202 .02(\(i+1\). Hence )J 473 323 :M -.385(B)A f0_6 sf 0 2 rm -.173(k)A 0 -2 rm f0_10 sf -.272(\(i\))A 59 335 :M .467 .047(is not a collider on B)J f0_6 sf 0 2 rm .121(k)A 0 -2 rm f0_10 sf .361 .036(, and not in )J f2_10 sf .269(Z)A f0_10 sf .091 .009( )J f1_10 sf .309A f0_10 sf .091 .009( )J f2_10 sf .325(S.)A 59 350 :M f0_10 sf .113 .011(Suppose that B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .076 .008(\(i\) is not a non-collider on )J f4_10 sf (U)S f0_10 sf .054 .005(. If k = 0, then B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .05 .005(\(i\) is in )J f2_10 sf (O)S f0_10 sf .07 .007( and an ancestor of )J f2_10 sf (Z)S f0_10 sf .084 .008(, and for every)J 59 362 :M (other )S 83 362 :M -.031(value )A 108 362 :M .144 .014(of )J 120 362 :M .833 .083(k, )J 132 362 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf ( )S 145 362 :M .694 .069(is )J 156 362 :M .056 .006(a )J 164 362 :M -.185(subsequence )A 216 362 :M .144 .014(of )J 228 362 :M .15(B)A f0_6 sf 0 2 rm .068(0)A 0 -2 rm f0_10 sf .102 .01(, )J 244 362 :M .509 .051(so )J 257 362 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 279 362 :M .694 .069(is )J 290 362 :M .601 .06(in )J 302 362 :M .601 .06(in )J 314 362 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 326 362 :M -.313(and )A 343 362 :M .051 .005(an )J 356 362 :M -.101(ancestor )A 392 362 :M .451(of)A f2_10 sf .271 .027( )J 405 362 :M .604(Z)A f0_10 sf .226 .023( )J 416 362 :M .328 .033(if )J 426 362 :M .786 .079(it )J 436 362 :M .694 .069(is )J 447 362 :M .555 .056(not )J 464 362 :M .056 .006(a )J 473 362 :M -.109(non-)A 59 374 :M .156 .016(collider on )J f4_10 sf .079(U)A f0_10 sf .133 .013(. We will now show )J 197 374 :M .361 .036(that )J 216 374 :M .218 .022(the )J 232 374 :M -.062(inducing )A 270 374 :M .263 .026(paths )J 295 374 :M -.116(between )A 331 374 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 353 374 :M -.313(and )A 370 374 :M .079(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .036(\(i)A f1_10 sf .065(-)A f0_10 sf .121 .012(1\), )J 406 374 :M -.313(and )A 423 374 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 445 374 :M -.313(and )A 462 374 :M -.167(B)A f0_6 sf 0 2 rm -.075(k)A 0 -2 rm f0_10 sf -.125(\(i+1\))A 59 386 :M -.235(are )A 75 386 :M .515 .052(both )J 98 386 :M .674 .067(into )J 119 386 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.015(\(i\). )A 145 386 :M -.188(We )A 163 386 :M -.094(have )A 186 386 :M -.275(already )A 218 386 :M .261 .026(shown )J 249 386 :M .361 .036(that )J 269 386 :M .218 .022(the )J 286 386 :M -.062(inducing )A 325 386 :M .202 .02(path )J 347 386 :M -.116(between )A 384 386 :M -.186(B)A f0_6 sf 0 2 rm -.084(0)A 0 -2 rm f0_10 sf -.111(\(i\) )A 407 386 :M -.313(and )A 425 386 :M -.05(B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 460 386 :M -.313(and )A 479 386 :M -.108(the)A 59 398 :M -.015(inducing path between B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.014(\(i\) and B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf -.013(1\) are into B)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.014(\(i\). Suppose the same is true for each B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.012(\(i\) that is )A 470 398 :M .555 .056(not )J 487 398 :M (a)S 59 410 :M -.114(non-collider )A 110 410 :M .417 .042(on )J 124 410 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 138 410 :M .517 .052(This )J 160 410 :M .676 .068(will )J 180 410 :M .281 .028(also )J 200 410 :M .051 .005(be )J 213 410 :M (true )S 232 410 :M -.052(for )A 247 410 :M -.204(each )A 268 410 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 290 410 :M .315 .032(unless )J 319 410 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 347 410 :M .694 .069(is )J 358 410 :M .051 .005(an )J 371 410 :M -.101(ancestor )A 407 410 :M .144 .014(of )J 419 410 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf (1\), )S 461 410 :M -.313(and )A 479 410 :M -.108(the)A 59 422 :M (vertex B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf -.003(1\) was removed at the k)A f0_6 sf 0 -4 rm (th)S 0 4 rm f0_10 sf ( step of the )S 262 422 :M .56 .056(algorithm, )J 308 422 :M .144 .014(or )J 320 422 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 348 422 :M .694 .069(is )J 359 422 :M .051 .005(an )J 372 422 :M -.101(ancestor )A 408 422 :M .144 .014(of )J 420 422 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i+1\), )S 462 422 :M -.313(and )A 479 422 :M -.108(the)A 59 434 :M .094 .009(vertex B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .07 .007(\(i+1\) was removed at the k)J f0_6 sf 0 -4 rm (th)S 0 4 rm f0_10 sf .072 .007( step of the algorithm. Suppose without loss of generality that B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i\))S 59 446 :M .694 .069(is )J 70 446 :M .051 .005(an )J 83 446 :M -.101(ancestor )A 119 446 :M .144 .014(of )J 132 446 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i+1\), )S 175 446 :M -.313(and )A 193 446 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 222 446 :M (was )S 242 446 :M -.212(removed )A 280 446 :M .236 .024(at )J 292 446 :M .218 .022(the )J 309 446 :M .342(k)A f0_6 sf 0 -4 rm .16(th)A 0 4 rm f0_10 sf .171 .017( )J 324 446 :M .281 .028(step )J 345 446 :M .144 .014(of )J 358 446 :M .218 .022(the )J 375 446 :M .56 .056(algorithm. )J 422 446 :M (The )S 442 446 :M -.344(edge )A 464 446 :M .417 .042(on )J 479 446 :M -.108(the)A 59 458 :M -.069(inducing path between B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm -.052(-)A 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i\) )A 179 458 :M -.313(and )A 196 458 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i+1\) )A 235 458 :M .361 .036(that )J 254 458 :M .098 .01(contains )J 291 458 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 319 458 :M .694 .069(is )J 330 458 :M .674 .067(into )J 350 458 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 378 458 :M -.313(and )A 395 458 :M .509 .051(so )J 408 458 :M .694 .069(is )J 419 458 :M -.141(every )A 444 458 :M -.344(edge )A 465 458 :M .417 .042(on )J 479 458 :M -.108(the)A 59 470 :M -.026(inducing path between B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.025(\(i+1\) and B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.023(\(i+2\) that contains B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.021(\(i\) \(because it is in )A f2_10 sf (O)S f0_10 sf -.022(.\) B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.023(\(i\) )A 435 470 :M .694 .069(is )J 446 470 :M .555 .056(not )J 463 470 :M .417 .042(on )J 477 470 :M -.219(any)A 59 482 :M -.029(directed path from B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.027(\(i+1\) to B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.028(\(i+2\) because by hypothesis, B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.025(\(i\) is an )A 365 482 :M -.101(ancestor )A 401 482 :M .144 .014(of )J 413 482 :M (B)S f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf (\(i+1\). )S 455 482 :M .19 .019(Since )J 481 482 :M (all)S 59 494 :M .144 .014(of )J 71 494 :M .218 .022(the )J 87 494 :M -.253(edges )A 112 494 :M .361 .036(that )J 131 494 :M .039 .004(contain )J 164 494 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 192 494 :M .417 .042(on )J 206 494 :M .218 .022(the )J 222 494 :M .263 .026(paths )J 247 494 :M -.207(used )A 268 494 :M .601 .06(to )J 280 494 :M -.017(construct )A 320 494 :M .218 .022(the )J 336 494 :M -.062(inducing )A 374 494 :M .202 .02(path )J 395 494 :M -.116(between )A 432 494 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 461 494 :M -.313(and )A 479 494 :M -.108(the)A 59 506 :M -.081(vertex )A 87 506 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i+2\) )A 126 506 :M -.235(are )A 141 506 :M .674 .067(into )J 161 506 :M -.087(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.047(\(i\), )A 192 506 :M .218 .022(the )J 208 506 :M -.062(inducing )A 246 506 :M .202 .02(path )J 267 506 :M -.116(between )A 303 506 :M -.214(B)A f0_6 sf 0 2 rm -.096(k)A 0 -2 rm f1_6 sf 0 2 rm -.106(-)A 0 -2 rm f0_6 sf 0 2 rm -.096(1)A 0 -2 rm f0_10 sf -.128(\(i\) )A 331 506 :M f1_10 sf .695A f0_10 sf .317 .032( )J 341 506 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 363 506 :M -.313(and )A 380 506 :M -.093(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f1_6 sf 0 2 rm (-)S 0 -2 rm f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf -.063(\(i+2\) )A 419 506 :M f1_10 sf .695A f0_10 sf .317 .032( )J 429 506 :M -.05(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.034(\(i+1\) )A 463 506 :M .694 .069(is )J 475 506 :M .148(into)A 59 518 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\).)A 59 533 :M .06 .006(If B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .063 .006(\(i\) is not a non-collider on )J f4_10 sf (U)S f0_10 sf .071 .007(, then the inducing paths )J 295 533 :M -.116(between )A 331 533 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 353 533 :M -.313(and )A 370 533 :M .079(B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .036(\(i)A f1_10 sf .065(-)A f0_10 sf .121 .012(1\), )J 406 533 :M -.313(and )A 423 533 :M -.186(B)A f0_6 sf 0 2 rm -.084(k)A 0 -2 rm f0_10 sf -.111(\(i\) )A 445 533 :M -.313(and )A 462 533 :M -.167(B)A f0_6 sf 0 2 rm -.075(k)A 0 -2 rm f0_10 sf -.125(\(i+1\))A 59 545 :M .156 .016(are both into B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .148 .015(\(i\). It follows that if the algorithm exits at stage k, each B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .141 .014(\(i\) that is not a non-collider on )J f4_10 sf (U)S 59 557 :M f0_10 sf .047 .005(is not an ancestor of either B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf (\(i)S f1_10 sf (-)S f0_10 sf .038 .004(1\) or B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .045 .005(\(i+1\). Hence it is a collider on B)J f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf .04 .004(, and is a member )J 428 557 :M .144 .014(of )J 440 557 :M f2_10 sf .546(O)A f0_10 sf .175 .018( )J 452 557 :M .361 .036(that )J 471 557 :M .694 .069(is )J 482 557 :M -.439(an)A 59 569 :M .05 .005(ancestor of )J f2_10 sf (Z)S f0_10 sf (.)S 59 584 :M -.03(Hence each vertex on B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.027( is active, and hence B)A f0_6 sf 0 2 rm (k)S 0 -2 rm f0_10 sf -.029( d-connects X and Y given )A f2_10 sf (Z)S f0_10 sf -.034( in MAG\()A f4_10 sf -.05(G)A f0_10 sf <28>S f2_10 sf -.054(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf -.02(\)\). )A f1_10 sf <5C>S 59 599 :M f2_10 sf .481 .048(Lemma 19:)J f0_10 sf .257 .026( If MAG\()J f4_10 sf .136(G)A f0_10 sf .063<28>A f2_10 sf .146(O)A f0_10 sf (,)S f2_10 sf .105(S)A f0_10 sf (,)S f2_10 sf .125(L)A f0_10 sf .249 .025(\)\) contains A *)J f1_10 sf .186A f0_10 sf .095 .01( B )J f1_10 sf .186A f0_10 sf .19 .019( H, and an edge A *)J f1_10 sf .188A f0_10 sf .255 .025(* H, then \(i\) the edge between)J 59 611 :M .16 .016(A and H is into H, and \(ii\) if )J 178 611 :M .255 .026(A )J 189 611 :M .111(*)A f1_10 sf .222A f0_10 sf .151 .015(* )J 213 611 :M .255 .026(H )J 224 611 :M .133 .013(has )J 241 611 :M .056 .006(a )J 249 611 :M -.213(different )A 285 611 :M .087 .009(orientation )J 332 611 :M .236 .024(at )J 343 611 :M .255 .026(A )J 354 611 :M .202 .02(than )J 375 611 :M .255 .026(A )J 386 611 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 405 611 :M .275 .028(B, )J 418 611 :M .202 .02(then )J 439 611 :M .218 .022(the )J 455 611 :M -.253(edges )A 480 611 :M -.602(are)A 59 623 :M -.02(oriented as A )A f1_10 sf -.05A f0_10 sf -.022( H and A )A f1_10 sf -.053A f0_10 sf -.03( B.)A 59 638 :M f2_10 sf .065(Proof.)A f0_10 sf .173 .017( Because there is an edge into H, A *)J f1_10 sf .148A f0_10 sf .158 .016(* H is not oriented as A *)J f1_10 sf .148A f0_10 sf .182 .018(o H. Because there is )J 450 638 :M .051 .005(an )J 463 638 :M -.344(edge )A 484 638 :M (A)S 59 650 :M (*)S f1_10 sf S f0_10 sf .005 0( B, B is not an ancestor of A. If the edge between A and H is oriented as A )J 376 650 :M f1_10 sf .504A f0_10 sf .128 .013( )J 390 650 :M .65 .065(H, )J 404 650 :M .202 .02(then )J 425 650 :M -.17(B )A 435 650 :M .694 .069(is )J 446 650 :M .051 .005(an )J 459 650 :M -.187(ancestor)A 59 662 :M .144 .014(of )J 71 662 :M .65 .065(A, )J 85 662 :M .043 .004(which )J 113 662 :M .694 .069(is )J 124 662 :M .056 .006(a )J 132 662 :M -.054(contradiction. )A 190 662 :M -.207(Hence )A 218 662 :M .218 .022(the )J 234 662 :M -.344(edge )A 255 662 :M -.116(between )A 291 662 :M .255 .026(A )J 302 662 :M -.313(and )A 319 662 :M .255 .026(H )J 330 662 :M .694 .069(is )J 341 662 :M .674 .067(into )J 361 662 :M .65 .065(H. )J 375 662 :M -.078(If )A 386 662 :M .218 .022(the )J 403 662 :M -.344(edge )A 425 662 :M .255 .026(A )J 437 662 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 457 662 :M .255 .026(H )J 469 662 :M .133 .013(has )J 487 662 :M (a)S 59 674 :M .108 .011(different orientation at A than A *)J f1_10 sf .074A f0_10 sf .076 .008( B, then either A )J f1_10 sf .078A f0_10 sf .059 .006( B and A )J f1_10 sf .074A f0_10 sf .054 .005( H, or A )J f1_10 sf .074A f0_10 sf ( )S 384 674 :M -.17(B )A 394 674 :M -.313(and )A 411 674 :M .255 .026(A )J 422 674 :M f1_10 sf .065A f0_10 sf ( )S 436 674 :M .65 .065(H. )J 450 674 :M -.078(If )A 460 674 :M .255 .026(A )J 471 674 :M f1_10 sf .504A f0_10 sf .128 .013( )J 485 674 :M (B)S 59 686 :M .031 .003(and A )J f1_10 sf S f0_10 sf .028 .003( H, then A is an ancest or of H \(A )J f1_10 sf S f0_10 sf ( B )S f1_10 sf S f0_10 sf .04 .004( H\) which contradicts A )J f1_10 sf S f0_10 sf ( H. )S f1_10 sf <5C>S endp %%Page: 11 11 %%BeginPageSetup initializepage (peter; page: 11 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 479 713 12 12 rC 479 722 :M f0_12 sf (11)S gR gS 0 0 552 730 rC 59 51 :M f2_10 sf .984 .098(Lemma 20:)J f0_10 sf .526 .053( If MAG\()J f4_10 sf .278(G)A f0_6 sf 0 2 rm .115(1)A 0 -2 rm f0_10 sf .128<28>A f2_10 sf .299(O)A f0_10 sf .096(,)A f2_10 sf .214(S)A f0_10 sf .096(,)A f2_10 sf .257(L)A f0_10 sf .654 .065(\)\) and MAG\()J f4_10 sf .278(G)A f0_6 sf 0 2 rm .115(2)A 0 -2 rm f0_10 sf .128<28>A f2_10 sf .299(O)A f0_10 sf .096(,)A f2_10 sf .171<53D5>A f0_10 sf .096(,)A f2_10 sf .192<4CD5>A f0_10 sf .424 .042(\)\) have the )J 339 51 :M (same )S 363 51 :M (basic )S 387 51 :M -.043(colliders, )A 427 51 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 438 51 :M .694 .069(is )J 449 51 :M .056 .006(a )J 457 51 :M .112(minimal)A 59 63 :M .16 .016(d-connecting path between X and Y given )J f2_10 sf .069(Z)A f0_10 sf .145 .014( in MAG\()J f4_10 sf .074(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .08(O)A f0_10 sf (,)S f2_10 sf .057(S)A f0_10 sf (,)S f2_10 sf .069(L)A f0_10 sf .101 .01(\)\), F is a collider on )J f4_10 sf .074(U)A f0_10 sf .096 .01(, H is an ancestor )J 483 63 :M -.328(of)A 59 75 :M f2_10 sf .175 .018(Z )J f0_10 sf .231 .023(and there is an F )J f1_10 sf .226A f0_10 sf .294 .029( H edge in MAG\()J f4_10 sf .166(G)A f0_6 sf 0 2 rm .069(1)A 0 -2 rm f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .127(S)A f0_10 sf .057(,)A f2_10 sf .153(L)A f0_10 sf .229 .023(\)\), then there is an F )J f1_10 sf .226A f0_10 sf .294 .029( H edge in MAG\()J f4_10 sf .166(G)A f0_6 sf 0 2 rm .069(2)A 0 -2 rm f0_10 sf .076<28>A f2_10 sf .178(O)A f0_10 sf .057(,)A f2_10 sf .102<53D5>A f0_10 sf .057(,)A f2_10 sf .115<4CD5>A f0_10 sf .105(\)\).)A 59 90 :M f2_10 sf .199(Proof.)A f0_10 sf .397 .04( If F is a collider on )J f4_10 sf .327(U)A f0_10 sf .573 .057(, by Lemma )J 233 90 :M .417 .042(16 )J 247 90 :M .515 .052(both )J 269 90 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 346 90 :M -.313(and )A 363 90 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 448 90 :M .039 .004(contain )J 481 90 :M -.155(X)A f0_6 sf 0 2 rm (0)S 0 -2 rm 59 102 :M f0_10 sf .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 78 102 :M .855 .086(F )J 88 102 :M f1_10 sf .359A f0_10 sf .248 .025(* )J 107 102 :M .37(Y)A f0_6 sf 0 2 rm .154(0)A 0 -2 rm f0_10 sf .233 .023(. )J 124 102 :M -.078(If )A 134 102 :M -.097(there )A 157 102 :M .694 .069(is )J 168 102 :M .417 .042(no )J 182 102 :M -.344(edge )A 203 102 :M -.116(between )A 239 102 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 253 102 :M -.313(and )A 270 102 :M .255 .026(H )J 281 102 :M .601 .06(in )J 293 102 :M .185(MAG\()A f4_10 sf .201(G)A f0_6 sf 0 2 rm .083(1)A 0 -2 rm f0_10 sf .092<28>A f2_10 sf .216(O)A f0_10 sf .069(,)A f2_10 sf .154(S)A f0_10 sf .069(,)A f2_10 sf .185(L)A f0_10 sf .249 .025(\)\), )J 373 102 :M .202 .02(then )J 394 102 :M .855 .086(F )J 404 102 :M .694 .069(is )J 415 102 :M .051 .005(an )J 428 102 :M -.182(unshielded )A 473 102 :M -.109(non-)A 59 114 :M .26 .026(collider on X)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .102 .01( *)J f1_10 sf .161A f0_10 sf .106 .011( F *)J f1_10 sf .163A f0_10 sf .215 .022(* H in MAG\()J f4_10 sf .118(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf .054<28>A f2_10 sf .127(O)A f0_10 sf (,)S f2_10 sf .091(S)A f0_10 sf (,)S f2_10 sf .109(L)A f0_10 sf .185 .019(\)\), and hence F )J 318 114 :M .694 .069(is )J 329 114 :M .051 .005(an )J 342 114 :M -.182(unshielded )A 387 114 :M -.114(non-collider )A 438 114 :M .417 .042(on )J 452 114 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 466 114 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 485 114 :M (F)S 59 126 :M .111(*)A f1_10 sf .222A f0_10 sf .151 .015(* )J 84 126 :M .255 .026(H )J 96 126 :M .601 .06(in )J 109 126 :M .278(MAG\()A f4_10 sf .301(G)A f0_6 sf 0 2 rm .125(2)A 0 -2 rm f0_10 sf .139<28>A f2_10 sf .324(O)A f0_10 sf .104(,)A f2_10 sf .185<53D5>A f0_10 sf .104(,)A f2_10 sf .208<4CD5>A f0_10 sf .374 .037(\)\). )J 198 126 :M .328 .033(It )J 210 126 :M .298 .03(follows )J 246 126 :M .361 .036(that )J 267 126 :M .855 .086(F )J 279 126 :M .111(*)A f1_10 sf .222A f0_10 sf .151 .015(* )J 305 126 :M .255 .026(H )J 318 126 :M .694 .069(is )J 331 126 :M -.158(oriented )A 368 126 :M .144 .014(as )J 382 126 :M .855 .086(F )J 394 126 :M .182(o)A f1_10 sf .359A f0_10 sf .091 .009( )J 415 126 :M .255 .026(H )J 428 126 :M .144 .014(or )J 442 126 :M .855 .086(F )J 454 126 :M f1_10 sf .504A f0_10 sf .128 .013( )J 470 126 :M .255 .026(H )J 483 126 :M .222(in)A 59 138 :M .099(MAG\()A f4_10 sf .107(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .115(O)A f0_10 sf (,)S f2_10 sf .066<53D5>A f0_10 sf (,)S f2_10 sf .074<4CD5>A f0_10 sf .155 .016(\)\). It is not oriented as F o)J f1_10 sf .146A f0_10 sf .163 .016( H because F is a collider on U and hence not an )J 447 138 :M -.101(ancestor )A 483 138 :M -.328(of)A 59 150 :M f2_10 sf .092(S)A f0_10 sf .187 .019(. It follows that the edge is oriented as F )J f1_10 sf .163A f0_10 sf .215 .021( H. Similarly, if there is no edge between Y)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .145 .014( and H, F *)J f1_10 sf .165A f0_10 sf .203 .02(* H)J 59 162 :M .173 .017(is oriented as F )J f1_10 sf .154A f0_10 sf .192 .019( H in MAG\()J f4_10 sf .113(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf .052<28>A f2_10 sf .122(O)A f0_10 sf (,)S f2_10 sf .069<53D5>A f0_10 sf (,)S f2_10 sf .078<4CD5>A f0_10 sf .229 .023(\)\). Suppose then that X)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .16 .016( and Y)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .2 .02( are both adjacent to H.)J 59 177 :M .007 .001(There is a vertex N on )J f4_10 sf (U)S f0_10 sf .007 .001( between X and F that is either \(i\) not adjacent to H, or \(ii\) the edge between )J 467 177 :M .255 .026(N )J 478 177 :M -.719(and)A 59 189 :M .255 .026(H )J 70 189 :M .694 .069(is )J 81 189 :M .555 .056(not )J 98 189 :M .674 .067(into )J 118 189 :M .65 .065(H, )J 132 189 :M .144 .014(or )J 144 189 :M .34 .034(\(iii\) )J 163 189 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 174 189 :M -.172(agrees )A 202 189 :M .517 .052(with )J 224 189 :M .218 .022(the )J 240 189 :M -.079(concatenation )A 298 189 :M .144 .014(of )J 310 189 :M f4_10 sf -.068(U)A f0_10 sf -.049(\(X,N\) )A 344 189 :M -.313(and )A 361 189 :M .218 .022(the )J 377 189 :M -.344(edge )A 398 189 :M -.116(between )A 434 189 :M .255 .026(N )J 445 189 :M -.313(and )A 462 189 :M .255 .026(H )J 473 189 :M .236 .024(at )J 484 189 :M (N)S 59 201 :M .148 .015(\(since X itself trivially satisifes condition \(iii\) )J 247 201 :M .328 .033(if )J 257 201 :M .786 .079(it )J 267 201 :M .694 .069(is )J 278 201 :M -.226(adjacent )A 313 201 :M .601 .06(to )J 325 201 :M .349 .035(H.\) )J 342 201 :M .891 .089(Similarly, )J 387 201 :M -.097(there )A 410 201 :M .694 .069(is )J 421 201 :M .056 .006(a )J 429 201 :M -.081(vertex )A 457 201 :M .555 .056(M )J 470 201 :M .417 .042(on )J 484 201 :M (U)S 59 213 :M .012 .001(between Y and F that is either \(i\) not adjacent to H, or \(ii\) the edge between M and H is )J 413 213 :M .555 .056(not )J 430 213 :M .674 .067(into )J 450 213 :M .65 .065(H, )J 464 213 :M .144 .014(or )J 476 213 :M (\(iii\))S 59 225 :M f4_10 sf (U)S f0_10 sf -.003( agrees with the concatenation of )A 204 225 :M f4_10 sf (U)S f0_10 sf (\(Y,M\) )S 240 225 :M -.313(and )A 257 225 :M .218 .022(the )J 273 225 :M -.344(edge )A 294 225 :M -.116(between )A 330 225 :M .555 .056(M )J 343 225 :M -.313(and )A 360 225 :M .255 .026(H )J 371 225 :M .236 .024(at )J 382 225 :M .926 .093(M. )J 398 225 :M .134 .013(Let )J 415 225 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 435 225 :M .051 .005(be )J 448 225 :M .218 .022(the )J 464 225 :M -.035(closest)A 59 237 :M .123 .012(such )J 81 237 :M -.081(vertex )A 109 237 :M .417 .042(on )J 123 237 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 134 237 :M .601 .06(to )J 146 237 :M 1.201 .12(F, )J 159 237 :M -.313(and )A 176 237 :M (Y)S f0_6 sf 0 2 rm -.017(o+1)A 0 -2 rm f0_10 sf ( )S 196 237 :M .051 .005(be )J 209 237 :M .218 .022(the )J 225 237 :M .169 .017(closest )J 256 237 :M .123 .012(such )J 278 237 :M -.081(vertex )A 306 237 :M .417 .042(on )J 320 237 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 331 237 :M .601 .06(to )J 343 237 :M 1.201 .12(F. )J 356 237 :M .134 .013(Let )J 373 237 :M .159(X)A f0_6 sf 0 2 rm .066(n)A 0 -2 rm f0_10 sf .055 .006( )J 387 237 :M .232 .023(through )J 422 237 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 436 237 :M .051 .005(be )J 449 237 :M .218 .022(the )J 465 237 :M .051 .005(be )J 479 237 :M -.108(the)A 59 249 :M -.02(vertices on )A f4_10 sf (U)S f0_10 sf -.024( between X)A f0_6 sf 0 2 rm (n+1)S 0 -2 rm f0_10 sf -.024( and F.)A 59 264 :M .239 .024(We will now show by induction that for 0 )J cF f1_10 sf .024A sf .239 .024( i )J cF f1_10 sf .024A sf .239 .024( n, the edge )J 302 264 :M -.116(between )A 338 264 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 351 264 :M -.313(and )A 368 264 :M .811 .081(its )J 382 264 :M -.09(successor )A 423 264 :M .417 .042(on )J 437 264 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 448 264 :M .694 .069(is )J 459 264 :M .674 .067(into )J 479 264 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf (,)S 59 276 :M .091 .009(and there is an edge X)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( )S f1_10 sf .072A f0_10 sf .09 .009( H in MAG\()J f4_10 sf .053(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .057(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf .073 .007(\)\). If X)J f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf .054 .005( = X)J f0_6 sf 0 2 rm .023(n+1)A 0 -2 rm f0_10 sf .091 .009(, then it is trivially true \(because there are )J 481 276 :M (no)S 59 288 :M -.072(vertices between X)A f0_6 sf 0 2 rm -.053(n+1)A 0 -2 rm f0_10 sf ( )S 147 288 :M -.313(and )A 164 288 :M .856 .086(F\). )J 180 288 :M .361 .036(Suppose )J 218 288 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 232 288 :M .694 .069(is )J 243 288 :M -.116(between )A 279 288 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 299 288 :M -.313(and )A 316 288 :M 1.201 .12(F. )J 329 288 :M -.188(We )A 346 288 :M -.094(have )A 368 288 :M -.275(already )A 399 288 :M .261 .026(shown )J 429 288 :M .361 .036(that )J 448 288 :M -.097(there )A 471 288 :M .694 .069(is )J 482 288 :M -.439(an)A 59 300 :M (edge X)S f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf ( *)S f1_10 sf S f0_10 sf ( F, and )S 139 300 :M .051 .005(an )J 152 300 :M -.344(edge )A 173 300 :M .855 .086(F )J 183 300 :M f1_10 sf .504A f0_10 sf .128 .013( )J 197 300 :M .65 .065(H. )J 211 300 :M -.085(By )A 226 300 :M -.038(definition )A 268 300 :M .144 .014(of )J 280 300 :M .133(X)A f0_6 sf 0 2 rm .057(n+1)A 0 -2 rm f0_10 sf .083 .008(, )J 303 300 :M .218 .022(the )J 319 300 :M -.344(edge )A 340 300 :M -.116(between )A 376 300 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 390 300 :M -.313(and )A 407 300 :M .255 .026(H )J 418 300 :M -.189(disagrees )A 457 300 :M .517 .052(with )J 479 300 :M -.108(the)A 59 312 :M -.008(edge between X)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.007( and F at X)A f0_6 sf 0 2 rm (0)S 0 -2 rm f0_10 sf -.007(. By Lemma 19 it follows that the edge between )A 369 312 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 383 312 :M -.313(and )A 400 312 :M .855 .086(F )J 410 312 :M .694 .069(is )J 421 312 :M .159(X)A f0_6 sf 0 2 rm .066(0)A 0 -2 rm f0_10 sf .055 .006( )J 435 312 :M f1_10 sf .504A f0_10 sf .128 .013( )J 449 312 :M 1.201 .12(F, )J 462 312 :M -.313(and )A 479 312 :M -.108(the)A 59 324 :M .381 .038(edge between X)J f0_6 sf 0 2 rm .06(0)A 0 -2 rm f0_10 sf .189 .019( and F is into X)J f0_6 sf 0 2 rm .06(0)A 0 -2 rm f0_10 sf .186 .019(. Suppose for 0 )J cF f1_10 sf .019A sf .186 .019( i )J cF f1_10 sf .019A sf .186 .019( )J 281 324 :M .462(m)A f1_10 sf .326(-)A f0_10 sf .405 .04(1 )J 304 324 :M .361 .036(that )J 323 324 :M .218 .022(the )J 339 324 :M -.344(edge )A 360 324 :M -.116(between )A 396 324 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 409 324 :M -.313(and )A 426 324 :M .811 .081(its )J 440 324 :M -.09(successor )A 481 324 :M (on)S 59 336 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 70 336 :M .694 .069(is )J 81 336 :M .674 .067(into )J 101 336 :M .579(X)A f0_6 sf 0 2 rm .134(i)A 0 -2 rm f0_10 sf .365 .036(, )J 117 336 :M -.313(and )A 134 336 :M .218 .022(the )J 150 336 :M -.344(edge )A 171 336 :M -.116(between )A 207 336 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 220 336 :M -.313(and )A 237 336 :M .255 .026(H )J 248 336 :M .694 .069(is )J 259 336 :M -.158(oriented )A 294 336 :M .144 .014(as )J 306 336 :M .389(X)A f0_6 sf 0 2 rm .09(i)A 0 -2 rm f0_10 sf .135 .013( )J 320 336 :M f1_10 sf .504A f0_10 sf .128 .013( )J 335 336 :M .65 .065(H. )J 350 336 :M -.078(If )A 361 336 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 378 336 :M .694 .069(is )J 390 336 :M -.116(between )A 427 336 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 448 336 :M -.313(and )A 466 336 :M .65 .065(H, )J 481 336 :M (by)S 59 348 :M -.038(definition )A 101 348 :M .144 .014(of )J 113 348 :M .133(X)A f0_6 sf 0 2 rm .057(n+1)A 0 -2 rm f0_10 sf .083 .008(, )J 136 348 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 152 348 :M .694 .069(is )J 163 348 :M -.226(adjacent )A 198 348 :M .601 .06(to )J 210 348 :M .255 .026(H )J 221 348 :M -.313(and )A 238 348 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 249 348 :M -.189(disagrees )A 288 348 :M .517 .052(with )J 310 348 :M .218 .022(the )J 326 348 :M -.079(concatenation )A 384 348 :M .144 .014(of )J 396 348 :M f4_10 sf (U)S f0_10 sf (\(X,X)S f0_6 sf 0 2 rm (m-1)S 0 -2 rm f0_10 sf (\) )S 440 348 :M -.313(and )A 457 348 :M .218 .022(the )J 474 348 :M -.626(edge)A 59 360 :M -.008(between X)A f0_6 sf 0 2 rm (m-1)S 0 -2 rm f0_10 sf -.007( and H at X)A f0_6 sf 0 2 rm (m-1)S 0 -2 rm f0_10 sf -.007(; hence the )A 214 360 :M -.344(edge )A 235 360 :M -.116(between )A 271 360 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 287 360 :M -.313(and )A 304 360 :M .23(X)A f0_6 sf 0 2 rm .103(m-1)A 0 -2 rm f0_10 sf .079 .008( )J 325 360 :M .694 .069(is )J 336 360 :M .308(X)A f0_6 sf 0 2 rm .199(m)A 0 -2 rm f0_10 sf .107 .011( )J 352 360 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 371 360 :M .369(X)A f0_6 sf 0 2 rm .164(m-1)A 0 -2 rm f0_10 sf .232 .023(. )J 395 360 :M -.085(By )A 410 360 :M .192 .019(lemma )J 441 360 :M .769 .077(19, )J 458 360 :M .218 .022(the )J 474 360 :M -.626(edge)A 59 372 :M -.011(between X)A f0_6 sf 0 2 rm (m)S 0 -2 rm f0_10 sf -.01( and X)A f0_6 sf 0 2 rm (m-1)S 0 -2 rm f0_10 sf -.009( is into X)A f0_6 sf 0 2 rm (m)S 0 -2 rm f0_10 sf -.009(, and there is an edge X)A f0_6 sf 0 2 rm (m)S 0 -2 rm f0_10 sf ( )S f1_10 sf S f0_10 sf -.01( H. Hence every vertex X)A f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf -.011( between X)A f0_6 sf 0 2 rm (n+1)S 0 -2 rm f0_10 sf ( and )S 474 372 :M .855 .086(F )J 484 372 :M .332(is)A 59 384 :M .023 .002(a collider on )J f4_10 sf (U)S f0_10 sf .022 .002(, and there is an edge X)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( )S f1_10 sf S f0_10 sf .029 .003( H. Similarly, every vertex Y)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf .03 .003( between Y)J f0_6 sf 0 2 rm (o+1)S 0 -2 rm f0_10 sf .015 .001( and F )J 429 384 :M .694 .069(is )J 440 384 :M .056 .006(a )J 448 384 :M -.13(collider )A 481 384 :M (on)S 59 396 :M f4_10 sf (U)S f0_10 sf .035 .004(, and there is an edge Y)J f0_6 sf 0 2 rm (i)S 0 -2 rm f0_10 sf ( )S f1_10 sf S f0_10 sf ( H.)S 59 411 :M .108 .011(If X)J f0_6 sf 0 2 rm .029(n+1)A 0 -2 rm f0_10 sf .113 .011( is adjacent to H, then by Lemma 19 X)J f0_6 sf 0 2 rm .029(n+1)A 0 -2 rm f0_10 sf .058 .006( *)J f1_10 sf .093A f0_10 sf .11 .011(* is into H, and by definition of X)J f0_6 sf 0 2 rm .029(n+1)A 0 -2 rm f0_10 sf (, )S f4_10 sf .067(U)A f0_10 sf .13 .013( agrees with the)J 59 423 :M -.089(concatenation of )A f4_10 sf -.158(U)A f0_10 sf -.111(\(X,X)A f0_6 sf 0 2 rm -.069(n+1)A 0 -2 rm f0_10 sf -.094(\) and the edge between )A 256 423 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 276 423 :M -.313(and )A 293 423 :M .255 .026(H )J 304 423 :M .236 .024(at )J 315 423 :M .133(X)A f0_6 sf 0 2 rm .057(n+1)A 0 -2 rm f0_10 sf .083 .008(. )J 338 423 :M .891 .089(Similarly, )J 383 423 :M .328 .033(if )J 393 423 :M -.078(If )A 403 423 :M (Y)S f0_6 sf 0 2 rm -.017(o+1)A 0 -2 rm f0_10 sf ( )S 423 423 :M .694 .069(is )J 434 423 :M -.226(adjacent )A 469 423 :M .601 .06(to )J 481 423 :M .281(H,)A 59 435 :M .202 .02(then )J 80 435 :M .417 .042(by )J 94 435 :M (Lemma )S 128 435 :M .417 .042(19 )J 142 435 :M (Y)S f0_6 sf 0 2 rm -.017(o+1)A 0 -2 rm f0_10 sf ( )S 162 435 :M .111(*)A f1_10 sf .222A f0_10 sf .151 .015(* )J 186 435 :M .694 .069(is )J 197 435 :M .674 .067(into )J 217 435 :M .65 .065(H, )J 231 435 :M -.313(and )A 248 435 :M .417 .042(by )J 263 435 :M -.038(definition )A 306 435 :M .144 .014(of )J 319 435 :M .133(Y)A f0_6 sf 0 2 rm .057(o+1)A 0 -2 rm f0_10 sf .083 .008(, )J 343 435 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 355 435 :M -.172(agrees )A 384 435 :M .517 .052(with )J 407 435 :M .218 .022(the )J 424 435 :M -.079(concatenation )A 483 435 :M -.328(of)A 59 447 :M f4_10 sf -.131(U)A f0_10 sf -.096(\(Y)A f0_6 sf 0 2 rm -.057(o+1)A 0 -2 rm f0_10 sf -.078(,Y\) and the )A 133 447 :M -.344(edge )A 154 447 :M -.116(between )A 190 447 :M (Y)S f0_6 sf 0 2 rm -.017(o+1)A 0 -2 rm f0_10 sf ( )S 210 447 :M -.313(and )A 227 447 :M .255 .026(H )J 238 447 :M .236 .024(at )J 249 447 :M .133(Y)A f0_6 sf 0 2 rm .057(o+1)A 0 -2 rm f0_10 sf .083 .008(. )J 272 447 :M .144 .014(In )J 284 447 :M .361 .036(that )J 303 447 :M -.041(case, )A 326 447 :M .218 .022(the )J 342 447 :M -.079(concatenation )A 400 447 :M .144 .014(of )J 412 447 :M f4_10 sf (U)S f0_10 sf -.022(\(X,X)A f0_6 sf 0 2 rm (n+1)S 0 -2 rm f0_10 sf (\), )S 458 447 :M .218 .022(the )J 474 447 :M -.626(edge)A 59 459 :M -.076(between X)A f0_6 sf 0 2 rm -.05(n+1)A 0 -2 rm f0_10 sf -.069( and H, the edge between H and Y)A f0_6 sf 0 2 rm -.05(o+1)A 0 -2 rm f0_10 sf -.071(, and U\(Y)A f0_6 sf 0 2 rm -.05(o+1)A 0 -2 rm f0_10 sf -.068(,Y\) d-connects X and Y given )A f2_10 sf -.107(Z)A f0_10 sf -.07(, and )A 453 459 :M .694 .069(is )J 464 459 :M -.127(shorter)A 59 471 :M .438 .044(than )J f4_10 sf .24(U)A f0_10 sf .151 .015(. )J 93 471 :M .517 .052(This )J 115 471 :M .694 .069(is )J 126 471 :M .056 .006(a )J 134 471 :M -.054(contradiction. )A 192 471 :M .328 .033(It )J 202 471 :M .298 .03(follows )J 236 471 :M .361 .036(that )J 255 471 :M -.044(either )A 281 471 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 301 471 :M .144 .014(or )J 313 471 :M (Y)S f0_6 sf 0 2 rm -.017(o+1)A 0 -2 rm f0_10 sf ( )S 333 471 :M .694 .069(is )J 344 471 :M .555 .056(not )J 361 471 :M -.226(adjacent )A 396 471 :M .601 .06(to )J 408 471 :M .65 .065(H. )J 422 471 :M .361 .036(Suppose )J 460 471 :M .074(without)A 59 483 :M .228 .023(loss of generality that it is the former.)J 59 498 :M -.078(If )A 69 498 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 90 498 :M .694 .069(is )J 102 498 :M .555 .056(not )J 120 498 :M -.226(adjacent )A 156 498 :M .601 .06(to )J 169 498 :M .255 .026(H )J 181 498 :M -.097(there )A 205 498 :M .694 .069(is )J 217 498 :M .056 .006(a )J 226 498 :M .202 .02(path )J 248 498 :M f4_10 sf .987(V)A f0_10 sf .404 .04( )J 260 498 :M -.116(between )A 297 498 :M (X)S f0_6 sf 0 2 rm -.017(n+1)A 0 -2 rm f0_10 sf ( )S 318 498 :M -.313(and )A 336 498 :M .255 .026(H )J 348 498 :M .474 .047(consisting )J 394 498 :M .144 .014(of )J 407 498 :M .218 .022(the )J 424 498 :M -.079(concatenation )A 483 498 :M -.328(of)A 59 510 :M f4_10 sf (U)S f0_10 sf (\(X)S f0_6 sf 0 2 rm (n+1)S 0 -2 rm f0_10 sf (,F\) )S 101 510 :M .517 .052(with )J 123 510 :M .218 .022(the )J 139 510 :M -.344(edge )A 160 510 :M -.116(between )A 196 510 :M .855 .086(F )J 206 510 :M -.313(and )A 223 510 :M .65 .065(H. )J 237 510 :M .134 .013(Let )J 254 510 :M f4_10 sf .421<56D5>A f0_10 sf .223 .022( )J 268 510 :M .051 .005(be )J 281 510 :M .218 .022(the )J 297 510 :M -.131(corresponding )A 356 510 :M .202 .02(path )J 378 510 :M .601 .06(in )J 391 510 :M .278(MAG\()A f4_10 sf .301(G)A f0_6 sf 0 2 rm .125(2)A 0 -2 rm f0_10 sf .139<28>A f2_10 sf .324(O)A f0_10 sf .104(,)A f2_10 sf .185<53D5>A f0_10 sf .104(,)A f2_10 sf .208<4CD5>A f0_10 sf .374 .037(\)\). )J 480 510 :M -.67(By)A 59 522 :M .057 .006(definition, )J 104 522 :M f4_10 sf .987(V)A f0_10 sf .404 .04( )J 115 522 :M .694 .069(is )J 126 522 :M .056 .006(a )J 134 522 :M -.019(discriminating )A 195 522 :M .202 .02(path )J 216 522 :M -.052(for )A 232 522 :M .855 .086(F )J 243 522 :M .601 .06(in )J 256 522 :M .185(MAG\()A f4_10 sf .201(G)A f0_6 sf 0 2 rm .083(1)A 0 -2 rm f0_10 sf .092<28>A f2_10 sf .216(O)A f0_10 sf .069(,)A f2_10 sf .154(S)A f0_10 sf .069(,)A f2_10 sf .185(L)A f0_10 sf .249 .025(\)\), )J 337 522 :M -.313(and )A 355 522 :M .855 .086(F )J 366 522 :M .694 .069(is )J 378 522 :M .056 .006(a )J 387 522 :M -.114(non-collider )A 439 522 :M .417 .042(on )J 454 522 :M .218 .022(the )J 471 522 :M .071(path.)A 59 534 :M -.259(Because )A 94 534 :M .388 .039(all )J 108 534 :M .144 .014(of )J 120 534 :M .218 .022(the )J 136 534 :M -.103(colliders )A 173 534 :M .417 .042(on )J 187 534 :M f4_10 sf .987(V)A f0_10 sf .404 .04( )J 198 534 :M -.235(are )A 213 534 :M .281 .028(also )J 233 534 :M -.103(colliders )A 270 534 :M .417 .042(on )J 284 534 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 295 534 :M .043 .004(which )J 324 534 :M .694 .069(is )J 336 534 :M .928 .093(minimal, )J 378 534 :M .417 .042(by )J 393 534 :M (Lemma )S 428 534 :M .417 .042(16 )J 443 534 :M .202 .02(they )J 465 534 :M -.235(are )A 481 534 :M (all)S 59 546 :M -.103(colliders )A 96 546 :M .417 .042(on )J 110 546 :M f4_10 sf .511<56D5>A f0_10 sf .492 .049(. )J 127 546 :M -.085(By )A 142 546 :M (Lemma )S 176 546 :M .769 .077(15, )J 193 546 :M f4_10 sf .421<56D5>A f0_10 sf .223 .022( )J 207 546 :M .694 .069(is )J 218 546 :M .056 .006(a )J 226 546 :M -.019(discriminating )A 288 546 :M .202 .02(path )J 310 546 :M -.052(for )A 326 546 :M 1.201 .12(F. )J 340 546 :M -.207(Hence )A 369 546 :M .855 .086(F )J 380 546 :M .694 .069(is )J 392 546 :M .056 .006(a )J 401 546 :M -.114(non-collider )A 453 546 :M .417 .042(on )J 468 546 :M f4_10 sf .421<56D5>A f0_10 sf .223 .022( )J 483 546 :M .222(in)A 59 558 :M .086(MAG\()A f4_10 sf .093(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .1(O)A f0_10 sf (,)S f2_10 sf .057<53D5>A f0_10 sf (,)S f2_10 sf .064<4CD5>A f0_10 sf .15 .015(\)\), and the edge between F and H is oriented as F )J f1_10 sf .127A f0_10 sf .158 .016( H in MAG\()J f4_10 sf .093(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .1(O)A f0_10 sf (,)S f2_10 sf .057<53D5>A f0_10 sf (,)S f2_10 sf .064<4CD5>A f0_10 sf .107 .011(\)\). )J f1_10 sf <5C>S 59 573 :M f2_10 sf .953 .095(Lemma 21)J f0_10 sf .537 .054(: If MAG\()J f4_10 sf .274(G)A f0_6 sf 0 2 rm .114(1)A 0 -2 rm f0_10 sf .126<28>A f2_10 sf .295(O)A f0_10 sf .095(,)A f2_10 sf .211(S)A f0_10 sf .095(,)A f2_10 sf .253(L)A f0_10 sf .645 .064(\)\) and MAG\()J f4_10 sf .274(G)A f0_6 sf 0 2 rm .114(2)A 0 -2 rm f0_10 sf .126<28>A f2_10 sf .295(O)A f0_10 sf .095(,)A f2_10 sf .168<53D5>A f0_10 sf .095(,)A f2_10 sf .189<4CD5>A f0_10 sf .428 .043(\)\) have )J 323 573 :M .218 .022(the )J 339 573 :M (same )S 363 573 :M (basic )S 387 573 :M -.043(colliders, )A 427 573 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 438 573 :M .694 .069(is )J 449 573 :M .056 .006(a )J 457 573 :M .112(minimal)A 59 585 :M .071 .007(d-connecting path between X and Y given )J f2_10 sf (Z)S f0_10 sf .065 .006( in MAG\()J f4_10 sf (G)S f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf .047 .005(\)\), A is a collider )J 388 585 :M .417 .042(on )J 402 585 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 413 585 :M -.313(and )A 430 585 :M -.17(B )A 440 585 :M .694 .069(is )J 451 585 :M .056 .006(a )J 459 585 :M -.152(member)A 59 597 :M .188 .019(of )J f2_10 sf .151(Z)A f0_10 sf .252 .025( that is the endpoint of a shortest path D from A to )J f2_10 sf .151(Z)A f0_10 sf .056 .006( )J 296 597 :M .601 .06(in )J 308 597 :M .185(MAG\()A f4_10 sf .201(G)A f0_6 sf 0 2 rm .083(1)A 0 -2 rm f0_10 sf .092<28>A f2_10 sf .216(O)A f0_10 sf .069(,)A f2_10 sf .154(S)A f0_10 sf .069(,)A f2_10 sf .185(L)A f0_10 sf .249 .025(\)\), )J 388 597 :M .202 .02(then )J 409 597 :M -.17(B )A 419 597 :M .694 .069(is )J 430 597 :M .056 .006(a )J 438 597 :M -.292(descendant )A 483 597 :M -.328(of)A 59 609 :M .634 .063(A in MAG\()J f4_10 sf .285(G)A f0_6 sf 0 2 rm .118(2)A 0 -2 rm f0_10 sf .131<28>A f2_10 sf .307(O)A f0_10 sf .099(,)A f2_10 sf .175<53D5>A f0_10 sf .099(,)A f2_10 sf .197<4CD5>A f0_10 sf .18(\)\).)A 59 624 :M f2_10 sf .115(Proof.)A f0_10 sf .348 .035( Let D\325 be the path corresponding to D in MAG\()J f4_10 sf .189(G)A f0_6 sf 0 2 rm .079(2)A 0 -2 rm f0_10 sf .087<28>A f2_10 sf .204(O)A f0_10 sf .065(,)A f2_10 sf .116<53D5>A f0_10 sf .065(,)A f2_10 sf .131<4CD5>A f0_10 sf .267 .027(\)\), and )J f4_10 sf .138<55D5>A f0_10 sf .206 .021( be the )J 403 624 :M .202 .02(path )J 424 624 :M -.131(corresponding )A 483 624 :M .222(to)A 59 636 :M f4_10 sf .164(U)A f0_10 sf .32 .032( in MAG\()J f4_10 sf .164(G)A f0_6 sf 0 2 rm .068(2)A 0 -2 rm f0_10 sf .076<28>A f2_10 sf .177(O)A f0_10 sf .057(,)A f2_10 sf .101<53D5>A f0_10 sf .057(,)A f2_10 sf .113<4CD5>A f0_10 sf .262 .026(\)\). By Lemma 16, A is a collider on )J f4_10 sf .12<55D5>A f0_10 sf .283 .028(. By Lemma 20, )J 382 636 :M .218 .022(the )J 398 636 :M .266 .027(first )J 418 636 :M -.344(edge )A 439 636 :M .417 .042(on )J 453 636 :M f4_10 sf <44D5>S f0_10 sf ( )S 467 636 :M .694 .069(is )J 478 636 :M .111(out)A 59 648 :M -.011(of A. If )A f4_10 sf <44D5>S f0_10 sf -.011( is not a directed )A 171 648 :M .202 .02(path )J 192 648 :M .601 .06(in )J 204 648 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 289 648 :M .202 .02(then )J 310 648 :M .786 .079(it )J 320 648 :M .098 .01(contains )J 357 648 :M .056 .006(a )J 365 648 :M -.13(collider )A 398 648 :M 1.201 .12(F. )J 411 648 :M -.259(Because )A 446 648 :M f4_10 sf .209(D)A f0_10 sf .072 .007( )J 457 648 :M -.207(does )A 478 648 :M .111(not)A 59 660 :M .039 .004(contain )J 92 660 :M .056 .006(a )J 100 660 :M -.06(collider, )A 136 660 :M -.313(and )A 153 660 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 230 660 :M -.313(and )A 247 660 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 333 660 :M -.094(have )A 356 660 :M .218 .022(the )J 373 660 :M (same )S 398 660 :M (basic )S 423 660 :M -.043(colliders, )A 464 660 :M .855 .086(F )J 475 660 :M .694 .069(is )J 487 660 :M (a)S 59 672 :M .261 .026(shielded collider in MAG\()J f4_10 sf .098(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .106(O)A f0_10 sf (,)S f2_10 sf .06<53D5>A f0_10 sf (,)S f2_10 sf .068<4CD5>A f0_10 sf .187 .019(\)\). It follows then that in MAG\()J f4_10 sf .098(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .106(O)A f0_10 sf (,)S f2_10 sf .075(S)A f0_10 sf (,)S f2_10 sf .09(L)A f0_10 sf .146 .015(\)\) there is a vertex )J 457 672 :M .356 .036(E )J 467 672 :M -.313(and )A 484 672 :M f4_10 sf (D)S 59 684 :M f0_10 sf -.01(contains a subpath E )A f1_10 sf S f0_10 sf ( F )S f1_10 sf S f0_10 sf -.01( H, and an edge between E and H. The edge between E and H is not oriented )A 483 684 :M -.328(as)A endp %%Page: 12 12 %%BeginPageSetup initializepage (peter; page: 12 of 12)setjob %%EndPageSetup gS 0 0 552 730 rC 479 713 12 12 rC 479 722 :M f0_12 sf (12)S gR gS 0 0 552 730 rC 59 51 :M f0_10 sf .096 .01(H )J f1_10 sf .117A f0_10 sf .161 .016( E, else MAG\()J f4_10 sf .086(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .092(O)A f0_10 sf (,)S f2_10 sf .066(S)A f0_10 sf (,)S f2_10 sf .079(L)A f0_10 sf .136 .014(\)\) contains a cycle; it is nor oriented as E )J f1_10 sf .124A f0_10 sf .125 .013( H because E )J 412 51 :M .694 .069(is )J 423 51 :M .051 .005(an )J 436 51 :M -.101(ancestor )A 472 51 :M .144 .014(of )J 484 51 :M (H)S 59 63 :M .633 .063(in MAG\()J f4_10 sf .21(G)A f0_6 sf 0 2 rm .087(1)A 0 -2 rm f0_10 sf .097<28>A f2_10 sf .227(O)A f0_10 sf .073(,)A f2_10 sf .162(S)A f0_10 sf .073(,)A f2_10 sf .194(L)A f0_10 sf .3 .03(\)\); it is neither E *)J f1_10 sf .291A f0_10 sf .247 .025(o H nor E )J 266 63 :M .111(o)A f1_10 sf .222A f0_10 sf .151 .015(* )J 290 63 :M .65 .065(H, )J 304 63 :M -.163(because )A 338 63 :M .255 .026(A )J 349 63 :M .202 .02(then )J 370 63 :M -.1(would )A 398 63 :M .051 .005(be )J 411 63 :M .051 .005(an )J 424 63 :M -.101(ancestor )A 460 63 :M .144 .014(of )J 472 63 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 483 63 :M .222(in)A 59 75 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .313 .031(\) )J 107 75 :M -.313(and )A 124 75 :M -.163(hence )A 150 75 :M .555 .056(not )J 167 75 :M .056 .006(a )J 175 75 :M -.13(collider )A 208 75 :M .417 .042(on )J 222 75 :M f4_10 sf .461(U)A f0_10 sf .29 .029(. )J 236 75 :M -.207(Hence )A 264 75 :M .786 .079(it )J 274 75 :M .694 .069(is )J 285 75 :M -.158(oriented )A 320 75 :M .144 .014(as )J 332 75 :M .356 .036(E )J 342 75 :M f1_10 sf .504A f0_10 sf .128 .013( )J 356 75 :M .65 .065(H. )J 370 75 :M .04 .004(But )J 388 75 :M .202 .02(then )J 409 75 :M f4_10 sf .209(D)A f0_10 sf .072 .007( )J 421 75 :M .694 .069(is )J 433 75 :M .555 .056(not )J 451 75 :M .056 .006(a )J 460 75 :M -.014(shortest)A 59 87 :M -.002(directed path from A to a member of )A f2_10 sf (Z)S f0_10 sf -.002(, contrary to our assumption. )A f1_10 sf <5C>S 59 102 :M f2_10 sf 2.722 .272(Theorem )J 110 102 :M 2.644 .264(1: )J 129 102 :M f0_10 sf (DAGs )S 162 102 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .313 .031(\) )J 214 102 :M -.313(and )A 235 102 :M f4_10 sf .561(G)A f0_6 sf 0 2 rm .233(2)A 0 -2 rm f0_10 sf .258<28>A f2_10 sf .604(O)A f0_10 sf .194(,)A f2_10 sf .345<53D5>A f0_10 sf .194(,)A f2_10 sf .388<4CD5>A f0_10 sf .411 .041(\) )J 296 102 :M -.235(are )A 316 102 :M -.16(d-separation )A 372 102 :M -.115(equivalent )A 421 102 :M .328 .033(if )J 436 102 :M -.313(and )A 458 102 :M .515 .052(only )J 485 102 :M -.106(if)A 59 114 :M .059(MAG\()A f4_10 sf .064(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .068(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .059(L)A f0_10 sf .15 .015(\)\) and MAG\()J f4_10 sf .064(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .068(O)A f0_10 sf (,)S f2_10 sf .039<53D5>A f0_10 sf (,)S f2_10 sf .044<4CD5>A f0_10 sf .138 .014(\)\) have the same basic colliders.)J 59 129 :M f2_10 sf .116(Proof.)A f0_10 sf .319 .032( If X and Y are adjacent in MAG\()J f4_10 sf .19(G)A f0_6 sf 0 2 rm .079(1)A 0 -2 rm f0_10 sf .088<28>A f2_10 sf .205(O)A f0_10 sf .066(,)A f2_10 sf .146(S)A f0_10 sf .066(,)A f2_10 sf .176(L)A f0_10 sf .248 .025(\)\) but not in )J 317 129 :M .278(MAG\()A f4_10 sf .301(G)A f0_6 sf 0 2 rm .125(2)A 0 -2 rm f0_10 sf .139<28>A f2_10 sf .324(O)A f0_10 sf .104(,)A f2_10 sf .185<53D5>A f0_10 sf .104(,)A f2_10 sf .208<4CD5>A f0_10 sf .374 .037(\)\), )J 405 129 :M .202 .02(then )J 426 129 :M -.052(for )A 441 129 :M .281 .028(some )J 466 129 :M (subset)S 59 141 :M f2_10 sf .951(V)A f0_10 sf .329 .033( )J 71 141 :M .144 .014(of )J 83 141 :M f2_10 sf .744(O)A f0_10 sf .435 .043(, )J 98 141 :M .255 .026(X )J 109 141 :M -.313(and )A 126 141 :M .255 .026(Y )J 137 141 :M -.235(are )A 152 141 :M -.325(d-separated )A 198 141 :M .189 .019(given )J 224 141 :M f2_10 sf .951(V)A f0_10 sf .329 .033( )J 236 141 :M f1_10 sf .62A f0_10 sf .202 .02( )J 248 141 :M f2_10 sf 1.339(S)A f0_10 sf .602 .06( )J 259 141 :M .601 .06(in )J 271 141 :M f4_10 sf .602(G)A f0_6 sf 0 2 rm .25(2)A 0 -2 rm f0_10 sf .277<28>A f2_10 sf .648(O)A f0_10 sf .208(,)A f2_10 sf .37<53D5>A f0_10 sf .208(,)A f2_10 sf .417<4CD5>A f0_10 sf .578 .058(\), )J 331 141 :M .555 .056(but )J 349 141 :M .555 .056(not )J 367 141 :M -.325(d-separated )A 414 141 :M .189 .019(given )J 441 141 :M f2_10 sf .951(V)A f0_10 sf .329 .033( )J 454 141 :M f1_10 sf .62A f0_10 sf .202 .02( )J 467 141 :M f2_10 sf 1.02<53D5>A f0_10 sf .574 .057( )J 483 141 :M .222(in)A 59 153 :M f4_10 sf .602(G)A f0_6 sf 0 2 rm .25(2)A 0 -2 rm f0_10 sf .277<28>A f2_10 sf .648(O)A f0_10 sf .208(,)A f2_10 sf .37<53D5>A f0_10 sf .208(,)A f2_10 sf .417<4CD5>A f0_10 sf .578 .058(\). )J 122 153 :M -.078(If )A 136 153 :M .755 .075(C )J 151 153 :M .182(*)A f1_10 sf .359A f0_10 sf .091 .009( )J 174 153 :M .855 .086(F )J 188 153 :M f1_10 sf .359A f0_10 sf .248 .025(* )J 211 153 :M .255 .026(D )J 226 153 :M .694 .069(is )J 241 153 :M .051 .005(an )J 258 153 :M -.182(unshielded )A 307 153 :M -.13(collider )A 344 153 :M .601 .06(in )J 360 153 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 441 153 :M .555 .056(but )J 462 153 :M .555 .056(not )J 483 153 :M .222(in)A 59 165 :M .04(MAG\()A f4_10 sf (G)S f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf .026<53D5>A f0_10 sf (,)S f2_10 sf .03<4CD5>A f0_10 sf .079 .008(\)\), then every set that d-separates C and )J 294 165 :M .255 .026(D )J 305 165 :M .601 .06(in )J 317 165 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 394 165 :M -.207(does )A 415 165 :M .555 .056(not )J 432 165 :M .039 .004(contain )J 465 165 :M 1.201 .12(F, )J 478 165 :M .111(but)A 59 177 :M -.141(every )A 84 177 :M .303 .03(set )J 99 177 :M .361 .036(that )J 118 177 :M -.224(d-separates )A 164 177 :M .755 .075(C )J 175 177 :M -.313(and )A 192 177 :M .255 .026(D )J 203 177 :M .601 .06(in )J 215 177 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 300 177 :M -.207(does )A 321 177 :M .039 .004(contain )J 354 177 :M 1.201 .12(F. )J 368 177 :M -.078(If )A 379 177 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 391 177 :M .694 .069(is )J 403 177 :M .056 .006(a )J 412 177 :M -.019(discriminating )A 474 177 :M -.072(path)A 59 189 :M .048 .005(between X and Y for F in MAG\()J f4_10 sf (G)S f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf (O)S f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf (L)S f0_10 sf .054 .005(\)\) and the corresponding path )J f4_10 sf <55D5>S f0_10 sf .043 .004( is a discriminating path for F )J 483 189 :M .222(in)A 59 201 :M .278(MAG\()A f4_10 sf .301(G)A f0_6 sf 0 2 rm .125(2)A 0 -2 rm f0_10 sf .139<28>A f2_10 sf .324(O)A f0_10 sf .104(,)A f2_10 sf .185<53D5>A f0_10 sf .104(,)A f2_10 sf .208<4CD5>A f0_10 sf .374 .037(\)\), )J 147 201 :M -.313(and )A 164 201 :M .855 .086(F )J 174 201 :M .694 .069(is )J 185 201 :M .056 .006(a )J 193 201 :M -.13(collider )A 226 201 :M .417 .042(on )J 240 201 :M f4_10 sf .209(U)A f0_10 sf .072 .007( )J 252 201 :M .555 .056(but )J 270 201 :M .555 .056(not )J 288 201 :M .417 .042(on )J 303 201 :M f4_10 sf .154<55D5>A f0_10 sf .132 .013(, )J 321 201 :M .202 .02(then )J 343 201 :M .417 .042(by )J 358 201 :M (Lemma )S 393 201 :M .454 .045(6 )J 403 201 :M -.097(there )A 427 201 :M .694 .069(is )J 439 201 :M .056 .006(a )J 448 201 :M .303 .03(set )J 464 201 :M f2_10 sf .604(Z)A f0_10 sf .226 .023( )J 476 201 :M (that)S 59 213 :M .265 .026(contains F that d-separates X and Y in MAG\()J f4_10 sf .129(G)A f0_6 sf 0 2 rm .054(2)A 0 -2 rm f0_10 sf .059<28>A f2_10 sf .139(O)A f0_10 sf (,)S f2_10 sf .079<53D5>A f0_10 sf (,)S f2_10 sf .089<4CD5>A f0_10 sf .236 .024(\)\) but not in MAG\()J f4_10 sf .129(G)A f0_6 sf 0 2 rm .054(1)A 0 -2 rm f0_10 sf .059<28>A f2_10 sf .139(O)A f0_10 sf (,)S f2_10 sf .099(S)A f0_10 sf (,)S f2_10 sf .119(L)A f0_10 sf .082(\)\).)A 59 228 :M .17 .017(If MAG\()J f4_10 sf .058(G)A f0_6 sf 0 2 rm (1)S 0 -2 rm f0_10 sf <28>S f2_10 sf .063(O)A f0_10 sf (,)S f2_10 sf (S)S f0_10 sf (,)S f2_10 sf .054(L)A f0_10 sf .137 .014(\)\) and MAG\()J f4_10 sf .058(G)A f0_6 sf 0 2 rm (2)S 0 -2 rm f0_10 sf <28>S f2_10 sf .063(O)A f0_10 sf (,)S f2_10 sf .036<53D5>A f0_10 sf (,)S f2_10 sf .04<4CD5>A f0_10 sf .111 .011(\)\) have the same basic colliders, then by Lemma 16 and )J 461 228 :M -.135(Lemma)A 59 240 :M .769 .077(21, )J 76 240 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 153 240 :M -.313(and )A 170 240 :M .245(MAG\()A f4_10 sf .265(G)A f0_6 sf 0 2 rm .11(2)A 0 -2 rm f0_10 sf .122<28>A f2_10 sf .286(O)A f0_10 sf .092(,)A f2_10 sf .163<53D5>A f0_10 sf .092(,)A f2_10 sf .184<4CD5>A f0_10 sf .28 .028(\)\) )J 255 240 :M -.094(have )A 277 240 :M .218 .022(the )J 293 240 :M (same )S 318 240 :M -.16(d-separation )A 370 240 :M .286 .029(relations. )J 412 240 :M -.085(By )A 428 240 :M (Lemma )S 463 240 :M .417 .042(17 )J 478 240 :M -.719(and)A 59 252 :M .33 .033(Lemma 18, )J 109 252 :M .255 .026(X )J 120 252 :M -.313(and )A 137 252 :M .255 .026(Y )J 148 252 :M -.235(are )A 163 252 :M -.325(d-separated )A 209 252 :M .189 .019(given )J 235 252 :M f2_10 sf .951(R)A f0_10 sf .329 .033( )J 247 252 :M .601 .06(in )J 259 252 :M .147(MAG\()A f4_10 sf .159(G)A f0_6 sf 0 2 rm .066(1)A 0 -2 rm f0_10 sf .073<28>A f2_10 sf .171(O)A f0_10 sf .055(,)A f2_10 sf .122(S)A f0_10 sf .055(,)A f2_10 sf .147(L)A f0_10 sf .168 .017(\)\) )J 336 252 :M .328 .033(if )J 346 252 :M -.313(and )A 363 252 :M .515 .052(only )J 385 252 :M .328 .033(if )J 395 252 :M .255 .026(X )J 406 252 :M -.313(and )A 423 252 :M .255 .026(Y )J 434 252 :M -.235(are )A 449 252 :M -.408(d-separated)A 59 264 :M .384 .038(given )J f2_10 sf .179(R)A f0_10 sf .056 .006( )J f1_10 sf .191A f0_10 sf .056 .006( )J f2_10 sf .138(S)A f0_10 sf .132 .013( in )J f4_10 sf .179(G)A f0_6 sf 0 2 rm .075(1)A 0 -2 rm f0_10 sf .083<28>A f2_10 sf .193(O)A f0_10 sf .062(,)A f2_10 sf .138(S)A f0_10 sf .062(,)A f2_10 sf .166(L)A f0_10 sf .35 .035(\). Similarly, X and Y are d-separated given )J f2_10 sf .179(R)A f0_10 sf .35 .035( in MAG\()J f4_10 sf .179(G)A f0_6 sf 0 2 rm .075(2)A 0 -2 rm f0_10 sf .083<28>A f2_10 sf .193(O)A f0_10 sf .062(,)A f2_10 sf .11<53D5>A f0_10 sf .062(,)A f2_10 sf .124<4CD5>A f0_10 sf .221 .022(\)\) if and )J 473 264 :M .074(only)A 59 276 :M .058 .006(if X and Y are d-separated given )J f2_10 sf (R)S f0_10 sf ( )S f1_10 sf S f0_10 sf ( )S f2_10 sf <53D5>S f0_10 sf ( )S 225 276 :M .601 .06(in )J 237 276 :M f4_10 sf .602(G)A f0_6 sf 0 2 rm .25(2)A 0 -2 rm f0_10 sf .277<28>A f2_10 sf .648(O)A f0_10 sf .208(,)A f2_10 sf .37<53D5>A f0_10 sf .208(,)A f2_10 sf .417<4CD5>A f0_10 sf .578 .058(\). )J 296 276 :M .328 .033(It )J 306 276 :M .298 .03(follows )J 340 276 :M .361 .036(that )J 359 276 :M f4_10 sf .426(G)A f0_6 sf 0 2 rm .177(1)A 0 -2 rm f0_10 sf .196<28>A f2_10 sf .459(O)A f0_10 sf .147(,)A f2_10 sf .328(S)A f0_10 sf .147(,)A f2_10 sf .393(L)A f0_10 sf .313 .031(\) )J 407 276 :M -.313(and )A 424 276 :M f4_10 sf .561(G)A f0_6 sf 0 2 rm .233(2)A 0 -2 rm f0_10 sf .258<28>A f2_10 sf .604(O)A f0_10 sf .194(,)A f2_10 sf .345<53D5>A f0_10 sf .194(,)A f2_10 sf .388<4CD5>A f0_10 sf .411 .041(\) )J 480 276 :M -.602(are)A 59 288 :M -.154(d-separation equivalent. )A f1_10 sf <5C>S 243 318 :M f2_10 sf .716(Bibliography)A 59 348 :M f0_10 sf .558 .056(Aho, )J 83 348 :M .985 .099(A., )J 100 348 :M -.01(Hopcroft, )A 142 348 :M 1.239 .124(J., )J 156 348 :M .317 .032(Ullman )J 190 348 :M .985 .099(D., )J 207 348 :M -.026(\(1974\) )A 237 348 :M (The )S 256 348 :M .109 .011(Design )J 288 348 :M -.313(and )A 305 348 :M .282 .028(Analysis )J 344 348 :M .144 .014(of )J 357 348 :M .282 .028(Computer )J 402 348 :M .691 .069(Algorithms, )J 456 348 :M -.316(Addison-)A 59 360 :M -.01(Wesley: Reading, MA.)A 59 375 :M .163 .016(Frydenberg, M., \(1990\), )J 160 375 :M (The )S 179 375 :M .409 .041(Chain )J 207 375 :M -.097(Graph )A 235 375 :M -.026(Markov )A 270 375 :M .299 .03(Property, )J 311 375 :M -.114(Scandinavian )A 367 375 :M .039 .004(Journal )J 400 375 :M .144 .014(of )J 412 375 :M .902 .09(Statistics, )J 456 375 :M .769 .077(17, )J 473 375 :M -.109(333-)A 59 387 :M .167(353.)A 59 402 :M -.035(Lauritzen, S., Dawid, A., Larsen, B., Leimer, H., \(1990\) Independence properties of directed Markov fields,)A 59 414 :M .317 .032(Networks, 20, 491-505.)J 59 429 :M .341 .034(Pearl, J., \(1988\). 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Association for Uncertainty in AI, Inc., Mountain View, CA.)J endp %%Trailer end %%EOF