Invariant1.Let(G,F,U,R)beaninstanc

Invariant1.Let(G,F,U,R)beaninstance.ForanyS-cycleCinG−UthatcontainsavertexofR,thereisanS-cycleCinGsuchthatV(C)=V(C)R.
Invariant1isclearlytruewhenRisempty.Wheneverwehideavertexv,wewillarguethattheinvariantisstilltrueaftervishidden.ThenextlemmashowsthatwecansafelyignoretheverticesinRwhenwemakefurtherdecisionsonG−U,andhenceitissafetoworkonG=G−(U∪R)insteadofG−U.
Lemma1.Let(G,F,U,R)beaninstance.UnderInvariant1,FisamaximalS-forestofG−UsuchthatF⊆FifandonlyifFRisamaximalS-forestofG.
Proof.LetFbeamaximalS-forestofG−UsuchthatF⊆F.ThenclearlyFRisanS-forestinG.Letusargueformaximality.SinceFismaximal,foranyvertexxofG−(U∪F),xisinvolvedinanS-cycleCinG−UsuchthatV(C)⊆F∪{x}.ObservethatsinceR⊆F⊆F,anysuchvertexxisalsoavertexinG.ByInvariant1,xisinvolvedinanS-cycleCinGsuchthatV(C)=V(C)R.SinceG=G−(U∪R),itfollowsthatV(C)⊆(FR)∪{x}.HencexcannotbeaddedtoFR,whichisthusamaximalS-forestofG.
Fortheotherdirection,assumethatFRisamaximalS-forestofG.HenceeveryvertexxinGoutsideofFRisinvolvedinanS-cycleCinGsuchthatV(C)⊆(FR)∪{x}.SinceG−UisasupergraphofG,CisalsoanS-cycleinG−U.HencenomoreverticescanbeaddedtoF,whichisthusmaximal.LetusarguethatFisanS-forest.Assumeforcontradictionthatitisnot.ThenavertexyofRisinvolvedinanS-cycleCinG−UsuchthatV(C)⊆F.ThenbyInvariant1,thereisanS-cycleCinGsuchthatV(C)⊆FR,whichcontradictstheassumptionthatFRisanS-forestofG.
Themeasureofaninstance(G,F,U,R)isthenumberofundecidedvertices,i.e.,theverticesinG−F.Inthebeginningofthealgorithmallverticesareundecidedandhencethemeasureof(G,∅,∅,∅)isn.ThemeasuredropsbythenumberofverticesdeletedfromGplusthenumberofverticesaddedtoF.Hidingavertexdoesnotaffectthemeasureofaninstance.Inthecallwithinput(G,F,U,R),thealgorithmwillfurtherbranchintosubproblemsinwhichsomeverticeswillbedeletedfromGandsomeverticeswillbeplacedinF,andthemeasurewilldropaccordingly.Ifatastep,webranchintotnewsubproblems,wherethemeasuredecreasesbyc1,c2,...,ctineachsubproblem,respectively,wegetthebranchingvector(c1,c2,...,ct).Ateachbranchingpoint,wewillgivethecorrespondingbranchingvectortobeofhelpintherunningtimeanalysis.
WenowdescribethereductionandthebranchingrulesofthealgorithmwhenGhasundecidedverticesandFisanS-forest.Let(G,F,U,R)beacallofthealgorithmsatisfyingthis.Inthebelow,weletN(v)=NG(v),N[v]=NG[v],andd(v)=dG(v).First,westatethreereductionrules.Theserulesareappliedrecursivelyontheconsideredinstanceaslongasitispossibletoapplyatleastoneofthem.
Itiseasytoseethatthefirstreductionrule,RuleA,issafesince{u,v,w}formsanS-cycle,andu,warealreadyplacedinF:
RuleA.IfinGanundecidedvertexvisadjacenttoverticesu,w∈Fsuchthatuw∈Eand{u,v,w}∩S
=∅,thendeletev,i.e.,reducetothesubproblem(G−v,F,U∪{v},R).
Thefollowingobservationimmediatelyresultsinthenextreductionrule:RuleB.
Observation1.LetvbeavertexofGsuchthatnoS-cycleofGcontainsv.ThenvmustbeaddedtoFifitisnotinF,anditisthensafetohidev.
Proof.IfvertexvisnotinvolvedinanS-cycle,thenitcannotgetinvolvedinanS-cycleatlaterstepswhenmoreandmoreverticesaredeletedfromGandaddedtoU.HenceitissafetoaddittoF,andduetomaximalityitmustbeaddedtoFifitisnotalreadyinF.Assumenowthatv∈F.RecallthatforanyS-cycleCinG−UthatcontainsavertexofR,thereisanS-cycleCinGsuchV(C)=V(C)R.BecausenoS-cycleinGcontainsv,CisanS-cycleinG−v.Hence,itissafetohidevandaddittoR.
RuleB.IfGhasavertexvwithd(v)1,thenaddvtoFifvisundecided,andwhenv∈Fthenhidev,i.e.,reducetothesubproblem(G−v,F∪{v},U,R∪{v}).
SinceGisnotemptyanditischordal,ithasasimplicialvertex.Withthefollowingobservationweobtainthenextreductionrule:RuleC.
Observation2.LetvbeasimplicialvertexofG.IfN[v]∩S=∅,thenvmustbeaddedtoFifitisnotalreadyinF,anditisthensafetohidev

ผลลัพธ์ (ไทย) 1:

Beaninstance Invariant1.Let (G, F, U, R)ForanyS-cycleCinG−UthatcontainsavertexofR, thereisanS-cycleC บริษัท suchthatV (C) = V (C) R.
Invariant1isclearlytruewhenRisempty.Wheneverwehideavertexv, wewillarguethattheinvariantisstilltrueaftervishiddenThenextlemmashowsthatwecansafelyignoretheverticesinRwhenwemakefurtherdecisionsonG−U, andhenceitissafetoworkonG = G− (U∪R) insteadofG−U.
Lemma1.Let (G, F, UR) beaninstanceUnderInvariant1, F isamaximalS forestofG−UsuchthatF⊆F ifandonlyifF RisamaximalS-forestofG.
Proof.LetF beamaximalS-forestofG−UsuchthatF⊆FThenclearlyF RisanS-forestinGIsmaximal Letusargueformaximality.SinceF, foranyvertexxofG− (U∪F), xisinvolvedinanS-cycleCinG−UsuchthatV (C) ⊆F ∪ {x }ObservethatsinceR⊆F⊆F, anysuchvertexxisalsoavertexinGByInvariant1suchthatV อิง xisinvolvedinanS-cycleC (C) = V (C) R.SinceG = G− (U∪R), itfollowsthatV (C) ⊆ (F R)∪{x }HencexcannotbeaddedtoF R,whichisthusamaximalS-forestofG.
Fortheotherdirection, assumethatF RisamaximalS-forestofGHenceeveryvertexxinG outsideofF RisinvolvedinanS-cycleCinG suchthatV (C) ⊆ (F R)∪{x }SinceG−UisasupergraphofG, CisalsoanS-cycleinG−U.HencenomoreverticescanbeaddedtoFwhichisthusmaximalLetusarguethatF isanS-forest.Assumeforcontradictionthatitisnot.ThenavertexyofRisinvolvedinanS-cycleCinG−UsuchthatV (C) ⊆FThenbyInvariant1, suchthatV (C) thereisanS-cycleC บริษัท ⊆F R,whichcontradictstheassumptionthatF RisanS-forestofG.
Themeasureofaninstance (G, F,U,R)isthenumberofundecidedvertices,i.e.,theverticesinG −FInthebeginningofthealgorithmallverticesareundecidedandhencethemeasureof (G ∅ ∅ ∅) ไม่PlusthenumberofverticesaddedtoF.Hidingavertexdoesnotaffectthemeasureofaninstance.Inthecallwithinput ThemeasuredropsbythenumberofverticesdeletedfromG (G, F, U, R), thealgorithmwillfurtherbranchintosubproblemsinwhichsomeverticeswillbedeletedfromG andsomeverticeswillbeplacedinF, andthemeasurewilldropaccordinglyIfatastep,webranchintotnewsubproblems,wherethemeasuredecreasesbyc1,c2,...,ctineachsubproblem,respectively,wegetthebranchingvector(c1,c2,...,ct)Ateachbranchingpoint, wewillgivethecorrespondingbranchingvectortobeofhelpintherunningtimeanalysis.
WenowdescribethereductionandthebranchingrulesofthealgorithmwhenG hasundecidedverticesandFisanS forest.Let(G,F,U,R) beacallofthealgorithmsatisfyingthisInthebelow, weletN (v) = NG (v), N [v] = NG [v], และ (v) =กิจ (v)Westatethreereductionrules แรกTheserulesareappliedrecursivelyontheconsideredinstanceaslongasitispossibletoapplyatleastoneofthem.
Itiseasytoseethatthefirstreductionrule, RuleA, issafesince {u, v, w } formsanS cycle,andu,warealreadyplacedinF:
RuleA.IfinG anundecidedvertexvisadjacenttoverticesu,w∈Fsuchthatuw∈Eand{u,v,w}∩S
=∅,thendeletev,i.e.,reducetothesubproblem(G −v,F,U∪{v},R).
Thefollowingobservationimmediatelyresultsinthenextreductionrule:RuleB.
Observation1.LetvbeavertexofG suchthatnoS cycleofG containsvThenvmustbeaddedtoFifitisnotinF, anditisthensafetohidev.
Proof.IfvertexvisnotinvolvedinanS-วงจร andaddedtoU.HenceitissafetoaddittoF thenitcannotgetinvolvedinanS-cycleatlaterstepswhenmoreandmoreverticesaredeletedfromGandduetomaximalityitmustbeaddedtoFifitisnotalreadyinF.Assumenowthatv∈F.RecallthatforanyS-cycleCinG−UthatcontainsavertexofR, thereisanS-cycleC บริษัท suchV (C) = V (C) R.BecausenoS-cycleinG containsv, −v isanS-cycleinG Cดังนั้น itissafetohidevandaddittoR.
RuleB.IfG hasavertexvwithd(v) 1,thenaddvtoFifvisundecided,andwhenv∈Fthenhidev,i.e.,reducetothesubproblem (G −v, F∪ {v } U, R∪ {v }) .
SinceG isnotemptyanditischordal, ithasasimplicialvertexWiththefollowingobservationweobtainthenextreductionrule:RuleC.
Observation2.LetvbeasimplicialvertexofGIfN [v] ∩S =∅ thenvmustbeaddedtoFifitisnotalreadyinF, anditisthensafetohidev

ผลลัพธ์ (ไทย) 3:

invariant1 ให้ (  G , F , U , r ) beaninstance . foranys cyclecing − uthatcontainsavertexofr thereisans cyclec ,  ไอเอ็นจี  suchthatv ( C  ) = V ( c ) R
invariant1isclearlytruewhenrisempty . wheneverwehideavertexv wewillarguethattheinvariantisstilltrueaftervishidden , . thenextlemmashowsthatwecansafelyignoretheverticesinrwhenwemakefurtherdecisionsong − U , andhenceitissafetoworkong  = G − ( − insteadofg ∪ R U ) U .
lemma1 . ให้  ( G , F , U ,R ) beaninstance . underinvariant1 F  isamaximals forestofg − usuchthatf ⊆ F  ifandonlyiff  risamaximals forestofg  .
หลักฐาน letf  beamaximals forestofg − usuchthatf ⊆ F  . thenclearlyf  risans foresting  . letusargueformaximality . sincef  ismaximal foranyvertexxofg − ( u , F ∪  ) xisinvolvedinans cyclecing − usuchthatv ( C ) ⊆ F  ∪ { x } observethatsincer ⊆ F ⊆ F  anysuchvertexxisalsoavertexing  byinvariant1 , , .xisinvolvedinans cyclec  ไอเอ็นจี  suchthatv ( C  ) = V ( c ) r.sinceg  = G − ( U ∪ R ( C ) , itfollowsthatv  ) ⊆ ( F  r ) ∪ { x } hencexcannotbeaddedtof  r , whichisthusamaximals forestofg  .
fortheotherdirection assumethatf  , N risamaximals forestofg henceeveryvertexxing  .  outsideoff  risinvolvedinans cyclecing  suchthatv ( C ) ⊆ ( F  r ) ∪ { x } sinceg − uisasupergraphofg  cisalsoans cycleing − u.hencenomoreverticescanbeaddedtof  , ,whichisthusmaximal . letusarguethatf  isans ป่า assumeforcontradictionthatitisnot . thenavertexyofrisinvolvedinans cyclecing − usuchthatv ( C ) ⊆ F  . thenbyinvariant1 thereisans cyclec ,  ไอเอ็นจี  suchthatv ( C  ) ⊆ F  r n whichcontradictstheassumptionthatf  risans forestofg  . 
themeasureofaninstance ( G  , F , U , r ) isthenumberofundecidedvertices theverticesing  ได้แก่ − Finthebeginningofthealgorithmallverticesareundecidedandhencethemeasureof ( G , ∅∅∅ , , ) มัน . themeasuredropsbythenumberofverticesdeletedfromg  plusthenumberofverticesaddedtof . hidingavertexdoesnotaffectthemeasureofaninstance . inthecallwithinput ( G  , F , U , r ) , thealgorithmwillfurtherbranchintosubproblemsinwhichsomeverticeswillbedeletedfromg  andsomeverticeswillbeplacedinf andthemeasurewilldropaccordingly , .ifatastep webranchintotnewsubproblems wherethemeasuredecreasesbyc1 C2 , , , , . . . , ctineachsubproblem ตามลำดับ wegetthebranchingvector ( C1 , C2 , . . . , CT ) ateachbranchingpoint wewillgivethecorrespondingbranchingvectortobeofhelpintherunningtimeanalysis , .
wenowdescribethereductionandthebranchingrulesofthealgorithmwheng  hasundecidedverticesandfisans ป่าให้ (  G , F , U , r ) beacallofthealgorithmsatisfyingthis .inthebelow weletn , ( v ) = ng  ( V ) [ V ] = ng  [ V ] andd ( V ) = DG  ( V ) ครั้งแรก westatethreereductionrules . theserulesareappliedrecursivelyontheconsideredinstanceaslongasitispossibletoapplyatleastoneofthem .
itiseasytoseethatthefirstreductionrule rulea issafesince { , , U , V , W } formsans รอบ andu warealreadyplacedinf :
, rulea . ifing  anundecidedvertexvisadjacenttoverticesu W ∈ fsuchthatuw ∈ eand { u , v , w } ∩ S
= ∅ thendeletev , คือreducetothesubproblem ( G  − V , F , u ∪ { v } , r )
thefollowingobservationimmediatelyresultsinthenextreductionrule : ruleb .
observation1 . letvbeavertexofg  suchthatnos cycleofg  containsv . thenvmustbeaddedtofifitisnotinf anditisthensafetohidev , .
หลักฐาน ifvertexvisnotinvolvedinans รอบ thenitcannotgetinvolvedinans cycleatlaterstepswhenmoreandmoreverticesaredeletedfromg andaddedtou.henceitissafetoaddittof  ,andduetomaximalityitmustbeaddedtofifitisnotalreadyinf . assumenowthatv ∈ f.recallthatforanys-cyclecing − uthatcontainsavertexofr thereisans cyclec ,  ไอเอ็นจี  suchv ( C  ) = V ( c ) r.becausenos-cycleing  containsv , C  isans cycleing  − v.hence itissafetohidevandaddittor , . 
ruleb . ifg  hasavertexvwithd ( V )  1 , thenaddvtofifvisundecided andwhenv ∈ , fthenhidev ได้แก่ reducetothesubproblem ( G  − V , F ∪ { v } , u , r ∪ { v } )
sinceg  isnotemptyanditischordal ithasasimplicialvertex , . withthefollowingobservationweobtainthenextreductionrule : rulec .
observation2 . letvbeasimplicialvertexofg  . IFN [ V ] ∩ S = ∅ thenvmustbeaddedtofifitisnotalreadyinf anditisthensafetohidev , ,