bmehu Abstract It is known to be NPhard to decide whether a graph can be made chordal by the deletion of vertices or by the deletion of edges Here we present a uniformly polynomialtime algorithm for b oth prob lems the running time is for some consta ID: 66980
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Chordaldeletionis xed-parametertractableDanielMarx?DepartmentofComputerScienceandInformationTheoryBudapestUniversityofTechnologyandEconomicsBudapestH-1521Hungarydmarx@cs.bme.huAbstract.ItisknowntobeNP-hardtodecidewhetheragraphcanbemadechordalbythedeletionofkverticesorbythedeletionofkedges.Herewepresentauniformlypolynomial-timealgorithmforbothprob-lems:therunningtimeisf(k)n forsomeconstant notdependingonkandsomefdependingonlyonk.Forlargevaluesofn,suchanalgo-rithmismuchbetterthantryingalltheO(nk)possibilities.Therefore,thechordaldeletionproblemparameterizedbythenumberkofverticesoredgestobedeletedis xed-parametertractable.ThisanswersanopenquestionofCai[4].1IntroductionAgraphischordalifitdoesnotcontainaninducedcycleoflengthgreaterthan3.Itcanbedecidedinlineartimewhetheragraphischordal[26].However,itisNP-completetodecidewhetheragraphcanbemadechordalbythedeletionofkvertices[19],bythedeletionofkedges[23],orbytheadditionofkedges[27](ifkispartoftheinput).Inthispaperweinvestigatetheseproblemsfromtheparameterizedcom-plexitypointofview.Parameterizedcomplexitydealswithproblemswheretheinputhasadistinguishedpartk(usuallyaninteger)calledtheparameter.Aparameterizedproblemiscalled xed-parametertractable(FPT)ifthereisanalgorithmwithrunningtimef(k)n ,wheref(k)isanarbitraryfunctionand isapositiveconstantindependentofk.ItturnsoutthatseveralNP-hardde-cisionproblems,suchasMinimumVertexCover(parameterizedbythesizekofthevertexcovertobefound)andLongestPath(parameterizedbythelengthkofthepath),are xed-parametertractable.Thefunctionf(k)isusu-allyexponential,thusiftheparameterkcanbearbitrary,thenthealgorithmsarenotpolynomial(asexpected).However,forsmall xedvaluesofk, xed-parametertractableproblemshavelow-degreepolynomialalgorithms,whicharesometimesevenpracticallyfeasible.Thede nitionof xed-parametertractabil-itycanbeextendedinastraightforwardwaytothecasewhentheinputhastwoparametersk1;k2.Inthiscase,ouraimisto ndanalgorithmwithrunningtime ?ResearchissupportedbytheMagyaryZoltanFels}ooktatasiKozalaptvanyandtheHungarianNationalResearchFund(OTKAgrant67651). f(k1;k2)n .Formorebackground,thereaderisreferredtothemonographofDowneyandFellows[8]ortotherecentbookofFlumandGrohe[9].Ifkisa xedconstant,thenthethreechordaldeletion/completionproblemscanbesolvedinpolynomialtimebyexhaustivesearch.Forexample,intheedgecompletionproblemwecantryallthenO(k)possibleedgesetsofsizekandcheckwhethertheadditionoftheseedgesmakesthegraphchordal.ThistrivialnO(k)timealgorithmcanbeimprovedtoO(4k=(k+1)3=2(n+m))time[3]orO(k2nm+k624k)time[16].Therefore,chordaledgecompletion(whichisalsocalledtheminimum ll-inproblem)is xed-parametertractable.Themainresultofthepaperisthatchordalvertexdeletionandchordaledgedeletionarealso xed-parametertractable.Infact,wegiveanalgorithmforthecommongeneralizationofthetwodeletionproblems:intheChordalDeletionproblemthegraphhastobemadechordalbythedeletionofatmostk1verticesandatmostk2edges.Theorem1.ChordalDeletionis xed-parametertractablewithcombinedparametersk1andk2,wherek1(resp.,k2)isthemaximumnumberofvertices(resp.,edges)tobedeleted.Cai[4]proposedageneralclassofgraphmodi cationproblemsanalogoustoChordalDeletion.LetGbeanarbitraryclassofgraphs.WedenotebyG+ke(resp.,G ke)theclassofthosegraphsthatcanbeobtainedbyadding(resp.,deleting)kedgesto/fromamemberofG.Similarly,letG+kvcontainthosegraphsthatcanbeobtainedfromsomememberofGbyaddingknewverticesandconnectingtheseverticeswiththeoriginalverticesandwitheachotherinanarbitraryway.(Anequivalentde nitionistosaythatagraphisinG+kvifitcanbemadeamemberofGbydeletingkvertices.)ForeverygraphclassG,wecanaskaboutthecomplexityofrecognizinggraphsinG+ke,G ke,orG+kv.Inparticular,weareinterestedinwhethertheseproblemsare xed-parametertractableparameterizedbyk.Ourmainresultimpliesthatrecognizingchordal+keandchordal+kvgraphsare xed-parametertractable.ThisanswersanopenquestionofCai[4].Theonlypreviousresultforthisproblemisalinear-timealgorithm[15]forrecognizingchordal+1eandchordal+1vgraphs,whichismoreecientthandeletingeachedge(vertex)andcheckingwhethertheremaininggraphischordal.Ouralgorithmcanactually ndthekedgesorkverticeswhosedeletionmakesthegraphchordal;theseedges/verticesarecalledthemodulatorofthegraphin[4].Vertexcoloringofchordal+kegraphsis xed-parametertractableparameterizedbyk,providedthatthemodulatorofthegraphisgivenintheinput[21].Theresultinthispaperimpliesthatthemodulatorofachordal+kegraphcanbegeneratedinf(k)n time,hencethevertexcoloringonchordal+kegraphsremains xed-parametertractableevenifthemodulatorisnotgivenintheinput.Theiterativecompressionmethodintroducedin[24]allowsustoconcentrateonaneasier\solutioncompression"problem.Thistechniqueprovedusefulformanyotherproblems,see[7,6,13].Thecompressionproblemisthefollowing(forbrevity,wediscussonlythevertex-deletionversioninthisparagraph):given2 asetXofk+1verticessuchthatGnXischordal, ndkverticeswhosedeletionmakesGchordal.Tosolvethissolutioncompressionproblem,we rstdeterminethesizeofthemaximumcliqueinthechordalgraphGnX.IfthecliquesizeGnXissmall,thenGnX(andhencetheslightlylargerG)hassmalltreewidth.Usingstandardtechniques,theproblemcanbesolvedinlineartimeforgraphswithboundedtreewidth.Ontheotherhand,weshowthatifthereisalargecliqueinGnX,thenthecliquecontains\irrelevant"verticesthatcanberemovedfromthegraphwithoutchangingthesolvabilityoftheproblem.Themaintechnicaldicultyoftheproofistoprovethatanirrelevantvertexalwaysexistsinalargeclique.Thisideaofrepeatedlydeletingirrelevantverticesuntilabounded-treewidthinstanceisobtainedwasusefulforotherproblemsaswell[25,12,22].Thepaperisorganizedasfollows.Section2reviewssomebasicfactsonchordalgraphs.Section3presentsthealgorithmforbounded-treewidthgraphs.InSection4weshowhowtheiterativecompressionmethodof[24]canbeappliedtoourproblem.Section5discusseshowwecanreducethesizeofthecliquestomakeourgraphaboundedtreewidthgraph.2ChordalgraphsWerecallsomestandardde nitionsfromgraphtheory.AwalkinagraphGisasequenceofverticesv1v2:::vksuchthatviandvi+1areadjacentinGforevery1ik.Thelengthofawalkv1v2:::vkisde nedtobek 1.Apathiswalkwherethevi'saredistinct.Wesaythatthepathv1v2:::vkconnectsverticesv1andvk.Thedistanceoftwoverticesuandvisthelengthoftheshortestpathconnectinguandv;thedistanceisde nedtobein nityifthereisnosuchpath.ThedistanceofavertexvandasetSofverticesistheminimumdistanceofvandavertexu2S.VertexvisadjacenttoSifthedistanceofvandSis1,i.e.,thereisanedgebetweenvandsomevertexu2S.AcycleinGisawalkv1v2:::vkvk+1suchthatv1=vk+1andvi=vjforevery1ijk.Thelengthofacyclev1v2:::vkvk+1isthenumberofdistinctverticesinthesequence,i.e.,k.Agraphischordalifitdoesnotcontainacycleoflengthgreaterthan3asaninducedsubgraph.Thisisequivalenttosayingthateverycycleoflengthgreaterthan3containsatleastonechord,i.e.,anedgeconnectingtwoverticesnotadjacentinthecycle.Achordlesscycleoflengthgreaterthan3willbecalledahole.Chordalityisahereditaryproperty:everyinducedsubgraphofachordalgraphischordal.Everychordalgraphisaperfectgraph[11]:theminimumnumberofcolorsrequiredtocolortheverticesofachordalgraphequalsthesizeofthelargestclique.Thecomplementofachordalgraphisalsoperfect,whichtranslatestothestatementthattheminimumnumberofcliquesrequiredtocovertheverticesofachordalgraphequalsthesizeofthelargestindependentset.Furthermore,anoptimumcoloringorcliquecoveringofachordalgraphcanbefoundin3 adeaefbfgabfbcgfedcbFig.1.Achordalgraphanditscliquetreedecomposition.polynomialtime[11].Wewillusetheseobservationstocovercertainsetsofverticeswithasmallnumberofcliquesandtreatthecliquesseparately.Chordalgraphscanbealsocharacterizedastheintersectiongraphsofsub-treesofatree(seee.g.,[11]):Theorem2.Thefollowingtwostatementsareequivalent:1.G(V;E)ischordal.2.ThereexistsatreeT(U;F)andasubtreeTvTforeachv2Vsuchthatu;v2VareneighborsinG(V;E)ifandonlyifTu\Tv=;(i.e.,TuandTvhaveacommonnode).ThetreeTtogetherwiththesubtreesTviscalledthecliquetreedecom-positionofG.Figure1showsachordalgraphandapossiblecliquetreede-composition.Theverticesinanodeofthetreeshowwhichsubtreescontainthatparticularnode;forexample,theleftmostnodeofthetreeiscontainedinsubtreesTbandTc.Onecan ndacliquetreedecompositionofagivenchordalgraphinpolynomialtime(see[11,26]).Forclarity,wewillusetheword\vertex"whenwerefertothegraphG(V;E),and\node"whenreferringtoT(U;F).WesaythatavertexvcoversnodexifTvcontainsnodex.ForanarbitrarynodexofT,theverticescoveringxinduceaclique.Conversely,foreverycliqueK,thereisanodexofTsuchthateveryv2Kcoversthisnodex(cf.[11]).Thefollowingeasyobservationwillbeusedrepeatedly:Proposition3.Letx,y,zbeverticesinG(V;E)suchthatxy;xz2Ebutyz62E.IfthereisawalkTinGnxfromytozsuchthatyandzaretheonlyneighborsofxinT,thenT[xcontainsaholeoflengthatleast4.Proof.LetPbeaminimalsubpathofTfromytoz.Sinceyandzarenotneighbors,pathPhaslengthatleast2.Therefore,thelengthofxyPzxisatleast4,anditischordless,sincePisaminimalpathandxisnottheneighboroftheinternalverticesofP.utProposition3canbealsothoughtofasacharacterizationofchordalgraphs:ifv1v2:::vtv1isahole,thenchoosingx=v1,y=v2,z=vtsatis estherequirements.IfthedeletionofXVandYEmakesthegraphG(V;E)chordal,thenwesaythatthepair(X;Y)isaholecoverofG.WeusethenotationGn(X;Y)4 forthegraphobtainedbydeletingtheverticesXandtheedgesYfromG.Thesizeofaholecover(X;Y)isthepair(jXj;jYj).Wesaythataholecover(X;Y)obstructsapathPifXcontainsavertexofPorYcontainsanedgeofP.Foraholecover(X;Y),let!(X;Y)containtheverticesofVandtheendpointsoftheedgesinE;clearlyj!(X;Y)jjXj+2jYj.Theproblemstudiedinthispaperisformallyde nedasfollows: ChordalDeletionInput:AgraphG(V;E)andintegersk1;k2Parameter:k1;k2Task:Fineaholecoverofsize(k1;k2). Itturnsoutthatthedeletionproblemisverydi erentfromtheedgecom-pletionproblem.Thealgorithmsin[3,16]forchordaledgecompletionusethestandardmethodofboundedsearchtrees.Ifthereisachordlesscycleoflengthmorethank+3,thentheanswerisno,aswewouldneedmorethankedgestomakethiscyclechordal.Ifthereisachordlesscycleoflength`k+3,theneverysolutionhastocontain` 3edgesthatmakethischordlesscyclechordal.Thereisaconstantnumberofdi erentwaysofmakingaholeofsize`chordalusing` 3edges.Thealgorithmtriesallthesepossibilities:webrancho intoatmostaconstant(i.e.,dependingonlyonk)numberofdirections.Aftermakingthecyclechordal,theproblemparameter(thenumberofedgesthatcanbeadded)isdecreasedby` 3,andthealgorithmcontinueswiththenextchordlesscycle.Sincetheproblemparametercanbedecreasedonlyatmostktimes,thealgorithm nishesafteratmostkbranchings.Ateachstep,thenumberofdirectionswebranchintocanbeboundedbyafunctionofk,thusthesizeofthesearchspaceexploredbythealgorithmcanbealsoboundedbyafunctionofk.Insummary,themainideaisthatthegraphcannotcontainalargehole,otherwisethegraphcouldnotbemadechordalbyaddingkedges.Inthedeletionproblemwecannotmakethisassumption:itispossiblethatthegraphcanbemadechordalbydeletingfewvertices,eveniftherearelargeholes(forexample,ifthegraphisalargechordlesscycle,thenitcanbemadechordalbythedeletionofasinglevertex).Thismeansthattheremightbemanypossi-bilitiestorepairalongchordlesscycle,thuswecannotusetheboundedsearchtreemethod.Substantiallydi erent(andmorecomplicated)ideasarerequiredforthevertexdeletionproblem.3Bounded-treewidthgraphsOnewaytode netreewidthisthefollowing:thetreewidthofagraphGisthesmallestintegerksuchthatGisasubgraphofachordalgraphHhavingcliquenumberk+1.Graphswithtreewidth1areexactlytheforests.Formorebackgroundontreewidth,seeforexample[18,2].ThealgorithmicimportanceoftreewidthcomesfromthefactthatalargenumberofNP-hardproblemscanbesolvedinlineartimeifwehaveabound5 onthetreewidthoftheinputgraph.Mostofthesealgorithmsuseabottom-updynamicprogrammingapproach,whichgeneralizesdynamicprogrammingontrees.Courcelle'sTheorem[5](seealso[8,Section6.5])givesapowerfulwayofquicklyshowingthataproblemislinear-timesolvableonboundedtreewidthgraphs.SentencesintheExtendedMonadicSecondOrderLogicofGraphs(EMSO)containquanti ers,logicalconnectives(:,_,and^),vertexvariables,edgevari-ables,vertexsetvariables,edgesetvariables,andthefollowingbinaryrelations:2,=,inc(e;v)(edgevariableeisincidenttovertexvariablev),andadj(u;v)(vertexvariablesu,vareneighbors).Ifagraphpropertycanbedescribedinthislanguage,thenthisdescriptioncanbeturnedintoanalgorithm:Theorem4(Courcelle[5]).IfagraphpropertycanbedescribedintheEx-tendedMonadicSecondOrderLogicofGraphs,thenforeveryw,thereisalinear-timealgorithmfortherecognitionofthispropertyongraphswithtreewidthatmostw.UsingProp.3,itisnotdiculttodescribethosegraphsG(V;E)thatcanbemadechordalbythedeletionofatmostk1verticesandatmostk2edges:(k1;k2)-chordal-deletion(V,E):=9v1;:::vk12V;9e1;:::;ek22E;V0V;E0E:chordal(V0;E0)^(8v2V:v2V0_v=v1__v=vk1)^(8e2E:e2E0_e=e1__e=ek2)chordal(V0;E0):=:(9x;y;z2V0;V1V0;E1E0:adj(x;y)^adj(x;z)^:adj(y;z)^(8q2V1:q=y_q=z_:adj(q;x))^connected(y;z;V1;E1))connected(y;z;V;E):=8Y;ZV:[(partition(V;Y;Z)^y2Y^z2Z)!(9y02Y;z02Z;e2E:inc(e;y0)^inc(e;z0))]partition(V;Y;Z):=8v2V:(v2Y_v2Z)^(v62Y_v62Z)Thepredicatechordal(V0;E0)expressesthatthesubgraphwithvertexsetV0andedgesetE0isachordalgraph.Totestwhetherthesubgraphischordal,wecheckwhetherthereareverticesx,y,andzsatisfyingtherequirementsofProp.3,i.e.,thereisatpathPwithverticesV1andedgesE1thatconnectyandzinsuchawaythattheinternalverticesarenotadjacenttox.ToensurethatyandzareconnectedbythepathP,werequirethatforeverypartitionY,ZofV1,ify2Yandz2Z,thenthereisanedgeofPconnectingYandZ.Courcelle'sTheoremtogetherwiththeEMSOformulationofChordalDele-tionimplies:Theorem5.Foreveryk1,k2,andw,ChordalDeletioncanbesolvedinlineartimeforgraphswithtreewidthatmostw.WenotethatTheorem5canbeobtainedwithoutCourcelle'sTheoremusingstandard(butverytediousandtechnical)dynamicprogrammingtechniques.6 ChordalDeletion(G;k1;k2)1.Seti:=k1andletXbetheverticesofGk1andYbek2arbitraryedges.2.Invariantcondition:(X;Y)isasize-(k1;k2)holecoverofGi.3.Ifi=n,thenreturn\(X;Y)isasize-(k1;k2)holecoverofG."4.SetX:=X[vi+1,now(X;Y)isasize-(k1+1;k2)holecoverofGi+1.5.CallHoleCoverCompression(Gi+1;k1;k2;X;Y).{Iftheanswerisasize-(k1;k2)holecover(X0;Y0)ofGi+1,thenlet(X;Y):=(X0;Y0),i:=i+1,andgotoStep2.{Iftheansweris\no,"thenreturn\no." Fig.2.AlgorithmChordalDeletion.4IterativecompressionReed,SmithandVetta[24]haveshownthattheBipartiteVertexDeletionproblem(makethegraphbipartitebydeletingkvertices)is xed-parametertractable.Theyintroducedthemethodofiterativecompression,whichcanbeusedinthecaseoftheChordalDeletionproblemaswell.Theideaisthatitissucienttoshowthatthefollowingeasierproblemis xed-parametertractable: HoleCoverCompressionInput:AgraphG,integersk1;k2,andaholecover(X;Y)ofsize(k1+1;k2).Parameter:k1;k2Task:Findaholecover(X0;Y0)ofsize(k1;k2)inG. ThisproblemiseasierthanChordalDeletion:theextrainput(X;Y)givesususefulstructuralinformationaboutG.Inparticular,weknowthatGn(X;Y)ischordal.Ouralgorithmbuildsheavilyonthisfact.Assumethatwehaveanalgorithmwithrunningtimef(k1;k2)n forHoleCoverCompression,thenChordalDeletioncanbesolvedasfollows(seeFigure2).Letv1,v2,:::,vnbeanorderingofthevertices,andletGibethegraphinducedbyv1,:::,vi.Wetryto ndasize-(k1;k2)holecoverforeachGi.GraphGk1triviallyhassuchaholecover.NowassumethatGihasasize-(k1;k2)holecover(X;Y).Clearly,(X[vi+1;Y)isasize-(k1+1;k2)holecoverofGi+1.Therefore,thecompressionalgorithmcanbeusedto ndasize-(k1;k2)holecoverforGi+1.Ifthereissuchaholecover,thenwecanproceedtoGi+2.Otherwisetheanswerisno,wecanconcludethatthesupergraphGofGi+1cannothaveasize-(k1;k2)holecovereither.Thealgorithmcallsthecompressionmethodatmostntimes,thusthetotalrunningtimeisf(k1;k2)n +1,whichshowsthattheproblemis xed-parametertractable.NotethatGi+1isobtainedfromGibyaddinganewvertex(ratherthananedge),thusthecompressionalgorithmisinvokedwithparameter(k1+1;k2)andnotwith(k1;k2+1).7 NowletusturnourattentiontotheHoleCoverCompressionalgorithmitself.Assumethatasize-(k1+1;k2)holecover(X;Y)ofGisgiven.LetW:=!(X;Y),letV0=VnW,anddenotebyG0thechordalgraphGnW.IfthesizeofthemaximumcliqueinV0isc,thenthetreewidthofthechordalgraphG0isc 1,andthetreewidthofGisatmostc 1+jWjc 1+k1+2k2+1.Therefore,ifthecliquesizeofG0canbeboundedbyaconstantdependingonk1andk2,thenthemethoddescribedforbounded-treewidthgraphsinSection3canbeusedtodecidewhetherGhasasize-(k1;k2)holecover.InSection5,wepresentamethodofreducingthecliquesizeofG0toaconstantdependingonlyonk1;k2.Avertexv2Visirrelevantifeverysize-(k1;k2)holecoverofGnvisalsoaholecoverofG.Ifweidentifyanirrelevantvertexv,thentheproblemcanbereducedto ndingasize-(k1;k2)holecoverinGnv.WeshowthatifthereisacliqueKinG0whosesizeisgreaterthansomeconstantck1;k2,thentheproblemcanbereducedtoasimplerform:eitherwe ndanirrelevantvertexorasmallsetofvertices/edgessuchthateverysize-(k1;k2)holecovercontainsatleastonememberofthisset.Moreprecisely,forasetNvofverticesandsetNeofedgeswesaythat(Nv;Ne)isanecessarysetifwhenever(X;Y)isasize-(k1;k2)holecover,theneitherXcontainsavertexofNvorYcontainsanedgeofNe.Iftheset(Nv;Ne)=(;;;)isanecessaryset,thenthismeansthatthereisnoholecoveroftherequiredsize.Thenecessarysetsthatwe ndarealwayssmall,i.e.,thereisaconstantbk1;k2suchthatjNvj+jNejbk1;k2.(Inthefollowing,whenwesay\anecessarysetcanbefound,"wealwaysmeanthatthesizeofthissetcanbeboundedbyafunctionofk1andk2.)Ifthecliquereductionalgorithmreturnsanecessaryset(Nv;Ne),thenwecanconcludethateverysize-(k1;k2)holecovercontainsatleastonevertexofNvoranedgeofNe.Therefore,webranchintojNvj+jNejdirections:foreachvertexvofNv,wecheckwhetherthereisasize-(k1 1;k2)holecoverofGnvandforeachedgeofNe,wecheckwhetherthereisasize-(k1;k2 1)holecover.Thustheproblemcanbereducedtoatmostbk1;k2subproblemswithsmallerparametervalues,wherebk1;k2dependsonlyonk.Insummary,thecliquereductionalgorithmdoesoneofthefollowing:{Identi esanirrelevantvertexv2K.Inthiscase,thedeletionofvdoesnotchangetheproblem.Ifthemaximumcliquesizeisstilllargerthanck1;k2,thenthealgorithmcanbeappliedagain.Otherwise,wecanusethealgorithmofTheorem5.{Identi esanecessaryset(Nv;Ne)whosesizeisboundedbyafunctionofk1andk2.Inthiscase,thealgorithmcanbranchintoaconstantnumberofdirections:onevertexofNvoroneedgeofNehastobedeleted.TheoverallalgorithmHoleCoverCompressionisshowninFigure3.Thealgorithmcallsthecliquereductionmethod(whichisdescribedinthefol-lowingsection)andcanmakesomenumberofrecursivecallstoHoleCoverCompressionwithparameter(k1 1;k2)andwithparameter(k1;k2 1).Thatis,thesumk1+k2strictlydecreasesineachrecursivecall,hencetherecursiondepthisatmostk1+k2.Byassumption,ifCliqueReductionreturnsanec-essaryset,thenitssizecanbeboundedbyafunctionofk1andk2.Thismeans8 HoleCoverCompression(G;k1;k2;X;Y)1.LetW:=!(X;Y).IfthecliquesizeofGnWisatmostck1;k2,thenusethealgorithmofTheorem5.2.IfGnWhasacliqueKofsizemorethanck1;k2,thencallCliqueRe-duction(G;W;K;k1;k2).3.Ifthereisanirrelevantvertexv,thendeletevfromG,andgotoStep1.4.Ifthereisanecessaryset(Nv;Ne):5.Foreachvertexv2Nv,callHoleCoverCompression(Gnv;k1 1;k2).{Iftheansweris\Yes"forsomev2Nv,and(X0;Y0)isasize-(k1 1;k2)holecoverofGnv,thenanswer\(X0[v;Y0)isasize-(k1;k2)holecoverofG."6.Foreachedgee2Ne,callHoleCoverCompression(Gne;k1;k2 1).{Iftheansweris\Yes"forsomee2Ne,and(X0;Y0)isasize-(k1;k2 1)holecoverofGne,thenanswer\(X0;Y0[e)isasize-(k1;k2)holecoverofG."7.Iftheansweris\No"foreveryvandeverye,thenanswer\No." Fig.3.AlgorithmHoleCoverCompression.thatthealgorithmbranchesintoaconstantnumberofdirections,andthesizeoftherecursiontreecanbealsoboundedbysomefunctionofk1andk2.ThustherunningtimeofHoleCoverCompressioncanbeboundedbyg(k1;k2)n foranappropriatefunctiongandconstant .5CliquereductionAsintheprevioussection,weassumethatWisasetofatmostk1+2k2+1verticessuchthatG0:=GnWisachordalgraph.InthissectionweshowthatifthereisalargecliqueKinG0,theninpolynomialtimewecaneither ndanecessarysetoranirrelevantvertexofK.Intherestofthesection,we xacliqueKinG0.Intuitivelyspeaking,avertexvofKisnotirrelevant,ifitissomehowessentialfortheholesofG.EveryholeofGgoesthroughavertexofW,thuseveryholeofGnotcompletelycontainedinWgoesthroughaneighborofWinG0.ThustheneighborsofWplayanimportantrole,hencewetrytounderstandthestructureofsuchverticesinSection5.1.ThoseneighborsofWareespeciallyimportantthatarereachablefromKincertaintechnicalsense,andhencecanbepartofaholecontainingalsoavertexofK.WewillinvestigatesuchverticesinSection5.2.ThesestructuralresultsenableustoidentifyaboundednumberofimportantverticesinthecliqueKandwecandeclareanyothervertexofthecliqueirrelevant(Section5.3).Moreprecisely,inSection5.4weshowthatifthereisaholegoingthroughsuchanirrelevantvertex(possiblyafterthedeletionofk1verticesandk2edges),thenthereisaholeavoidingthisvertex.Thisshowsthatremovingtheirrelevantvertexdoesnotchangetheanswertotheproblem.9 5.1LabelingIfavertexv2VnWistheneighborofsomevertex`2W,thenwesaythatvhaslabel`.Avertexcanhavemorethanonelabel;thelabelsofagivenvertexformasubsetofW.Thefollowingeasyobservationswillbeusedto ndnecessarysetsifcertainstructuresappearinthegraphG0:Proposition6.IfPisapathoflengthatleast2connectinguandv,andverticesuandvaretheonlyverticesinPhavinglabel`,theneveryholecoverhastocontaineither`,`u,`voratleastonevertexoredgeofP.Proof.If(X;Y)isaholecoverdisjointfromPandcontainsnoneofvertex`,edges`u,and`v,then`uPv`containsaholeinGn(X;Y)(Prop.3),acontradiction.utLemma7.Letvbeavertexwithoutlabelt,letx1,:::,xk1+k2+2beindependentt-labeledvertices,andletP1,:::,Pk1+k2+2beinternallydisjointpathswherePiconnectsvandxi,andtheinternalverticesofPidonothavelabelt.Then(fv;tg;;)isanecessaryset.Proof.Let(X;Y)beaholecoverofsize-(k1;k2)disjointfrom(fv;tg;;).Con-sidertheinternallydisjointpathsvPixitforeveryi=1;:::;k1+k2+2.Sincev;t62X,holecover(X;Y)canobstructatmostk1+k2ofthesepaths.AssumewithoutlossofgeneralitythatvP1x1tandvP2x2tarenotobstructed;thismeansthatx1andx2canbeconnectedwithapathx1P1vP2x2whoseinternalverticesdonothavelabelt.Sincex1andx2areneighborsoftinGn(X;Y)andthereisnoedgebetweenthem,Prop.6impliesthatthereisaholeinGn(X;Y).Lemma8.LetH1,:::,Hk1+k2+1beholesinG,letSbethesetofallverticesthatarecontainedinmorethanoneHi,andletESbetheedgesinducedbyS.IfjSjcforsomeconstantcdependingonlyonk1andk2,then(S;ES)isanecessarysetofsizeatmostc+c(c 1)=2.Proof.Let(X;Y)beaholecoverofsize-(k1;k2)suchthatS\X=;andSE\Y=;.NoweachvertexofXandeachedgeofYcanbecontainedinatmostoneholeHi.Thustherehastobeaholewhichisnotcoveredby(X;Y),acontradiction.InLemma10wegiveaboundonthenumberofindependentlabeledverticesintheneighborhoodofaconnectedunlabeledset.WeneedthefollowinglemmaofKleinberg[17]:Lemma9(Kleinberg[17]).LetAbeasetofvertices.Supposethatforsomek,theredonotexistsk+1pairwisedisjointpathswithdistinctendpointsinA.ThenthereisasetZofsizeatmost3ksuchthateachcomponentofGnZcontainsatmostonevertexofAnZ.Notethatthereisapolynomial-timealgorithmthat ndsk+1pairwisedisjointpathswithdistinctendpointsinA(ifsuchpathsexist)[10]andtheproofofLemma9canbemadealgorithmic.Thusinpolynomialtimewecaneither ndthek+1disjointpathsorthesetZofsize3k.10 Lemma10.LetBbeaconnectedsubsetofV0=V(G0)suchthatnovertexinBhaslabelt.LetIbeanindependentsetoft-labeledverticesintheneighborhoodofB.IfjIj�6(k1+k2)2,thenwecan ndanecessarysetinpolynomialtime.Proof.LetI=fv1;v2;:::;v6(k1+k2)2+1gbeanindependentsetofverticeswithlabeltintheneighborhoodofB.DenotebyG00thesubgraphofG0inducedbyI[B.Iftherearek1+k2+1disjointpathsinG00withdistinctendpointsinI,thenthesepathstogetherwithvertextgivek1+k2+1holesthatintersectonlyinvertext.ByLemma8,thismeansthatwecan ndanecessaryset.Assumethereforethattherearenosuchpaths;byLemma9,thismeansthatthereisasetZofsizeatmost3k1+3k2suchthateachcomponentofG00nZcontainsatmostonevertexofI.LetC1,:::,CcbethecomponentsofG00nZcontainingavertexofI,andletvibetheuniquevertexCi\I.NotethatcjInZj6(k1+k2)2+1 3(k1+k2)�3(k1+k2)(k1+k2+1)(ifk1+k2�1).WeclaimthateachCiisadjacenttoavertexofZ\B.First,itisnotpossiblethatZ\B=;:verticesviandvjareintheneighborhoodofB,hencetheycanbeconnectedwithapathwhoseinternalverticesareinB,andthispathwouldnotbeblockedbyZifB\Z=;.Letz2B\Zbeanarbitraryvertex.Eachvertexvihasaneighboru2B.Ifu2Z,thenuisaneighborofCiinZ\B.Otherwise,thereisapathfullycontainedinBthatconnectsuandz.Letz0bethe rstvertex(startingfromu)onthispaththatisinZ.Nowz0isaneighborofCi.SincejZ\Bj3(k1+k2),therehastobeavertexz2Z\Bthatisadjacenttomorethank1+k2+1components.AssumewithoutlossofgeneralitythatzisadjacenttocomponentsC1,:::,Ck1+k2+2,andpathPiconnectsvertexviwithzsuchthattheinternalverticesofPiareinCi.NotethatthesepathsintersectonlyinZ\B.Sincez2Z\Bdoesnothavelabelt,Lemma7givesanecessaryset.ut5.2DangerousverticesLetus xamaximalcliqueKofG0.Avertexv2V0nKiscalledat-dangerousvertex(forK)ifvhaslabeltandthereisapathPfromvtoavertexu2Ksuchthatvistheonlyvertexhavinglabeltonthepath.Vertexvisat-dangerousvertexifvhaslabeltandthereisapathPfromvtoavertexu2Ksuchthatvanduarenotneighbors,ualsohaslabelt,andtheinternalverticesofthepathdonothavelabelt.Vertexuisat-witness(t-witness)ofv,thepathPisat-witness(t-witness)pathofv.Avertexvcanbet-dangerousformorethanonet2W,oritcanbet-andt-dangerousatthesametime.ForasubgraphG00ofG0,weusetheexpressionwithrespecttoG00ifwerequirethatthewitnesspathisinG00.ThenamedangerouscomesfromtheobservationthatifthereisaholeinGthatgoesthroughthecliqueK,thentheholehastogothroughadangerousvertexaswell.Forexample,ifaholestartsint2W,goestoat-labeledneighborv2V0nKoft,goestoat-labeledvertexu2KviaapathPV0,andreturnstot,thenvisat-dangerousvertex,uisitswitness,andPisthewitnesspath11 (a)(b)v2v1t2t1 P KKtvuuFig.4.(a)At-dangerousvertexv.(b)At1-dangerousvertexv1andat2-dangerousvertexv2.(seeFigure4a).InthesituationdepictedinFigure4b,theholegoesthroughtwoverticest1;t2ofW,andtheholehasasubpathwithendpointsv1;v2thatgoesthroughK(wherev1andv2aretheneighborsoft1andt2,respectively).Theinternalverticesofthispathdonothavelabelst1;t2,hencev1ist1-dangerousandv2ist2-dangerous,anduisawitnessforboth.Whenwedeleteverticestomakethegraphchordal,ouraimistodestroyasmanywitnesspathsaspossibleandtomakemanyverticesnon-dangerous.Itwillturnoutthatifacliqueislarge,thenitcontainsmanyverticeswhosedeletiondoesnota ectthedangerousvertices,thusthereisnouseofdeletingthem.Weprovetwotechnicalresultsondangerousvertices:weboundby6(k1+k2)2(resp.,6(k1+k2)3)thenumberofindependentt-dangerous(resp.,t-dangerous)vertices.SinceG0ischordal(henceperfect),itfollowsthattheseverticescanbecoveredby6(k1+k2)2(resp.,6(k1+k2)3)cliques.Lemma11.GivenasetIofmorethan6(k1+k2)2independentt-dangerousvertices,wecan ndanecessarysetinpolynomialtime.Proof.ConsiderthesubgraphG00ofG0inducedbythoseverticesthatdonothavelabelt.ThecliqueKcontainsverticesonlyfromoneconnectedcomponentofG00,letBbethiscomponent.Clearly,everyt-dangerousvertexisaneighborofBinG0.Therefore,byLemma10,wecan ndanecessaryset.utLemma12.GivenasetIofmorethan6(k1+k2)3independentt-dangerousvertices,wecan ndanecessarysetinpolynomialtime.Proof.ConsiderthesubgraphG00ofG0inducedbytheverticeswithoutlabelt.LetC1,:::,CcbetheconnectedcomponentsofG00.Theinternalverticesofawitnesspathforat-dangerousvertexarecompletelycontainedinoneofthesecomponents.LetIiIcontainat-dangerousvertexv2IifandonlyifvhasawitnesspathwithinternalverticesonlyinCi.IfjIij�6(k1+k2)2forsome1ic,thenwearereadybyusingLemma10fortheconnectedsubgraphCi.Thusc�k1+k2,otherwisethesizeofthe12 independentsetisatmost6(k1+k2)3.Letus xk1+k2+1ofthesecomponents.ForeachsuchcomponentCi,letusselectat-dangerousvertexthathasawitnesspathPiwhoseinternalverticesareinCi.EachpathPitogetherwithvertextformahole.AstheinternalverticesofthePi'sareindi erentcomponents,thek1+k2+1holescanintersecteachotheronlyintheirendpointsandint.Thismeansthatthereareonly2k1+2k2+3verticesthatarecontainedinmorethanoneoftheholes;therefore,byLemma8,wecan ndanecessarysetofboundedsize.ut5.3MarkingthecliqueInthenexttwolemmas,weshowthatforacliqueQofdangerousvertices,thereisonlyaconstant(i.e.,dependingonlyonk1;k2)numberofverticesinKwhosedeletioncanmakeadangerousvertexofQnon-dangerous.Foreveryothervertexu2K,ifvist-dangerous,thenv2Qremainst-dangerouswithrespecttoG0nu.Evenmoreistrue:ifXisasetofatmostk1verticesandYisasetofatmostk2edges,thenv2Qist-dangerouswithrespecttoG0n(X;Y)ifandonlyifvist-dangerouswithrespecttoG0n(X[u;Y).Inthefollowinglemma,wemarksomenumberofverticessuchthatanyunmarkedvertexu2Khasthisproperty.Essentially,wehavetomarkthoseverticesofKthatare\closest"toQ,whereclosenessismeasuredinthecliquetreedecomposition.Lemma13.LetQbeacliqueoft-dangerousvertices.Foreveryk1;k2,thereisaconstantdk1;k2,suchthatwecanmarkdk1;k2verticesinKsuchthatifXisasetofk1vertices,andYisasetofk2edges,andv2Qhasanunmarkedt-witnessuwithrespecttoG0n(X;Y),thenvhasamarkedt-witnessu02Kn!(X;Y)withrespecttoG0n(X[u;Y).Proof.ConsiderthecliquetreedecompositionofthechordalgraphG0.SinceQandKarecliques,therearetwonodesxandysuchthateveryvertexofQcoversnodex,andeveryvertexofKcoversnodey.ConsiderthoseverticesofKthatdonothavelabelt,andordertheseverticessuchthatthedistanceoftheirsubtreesfromnodexisnondecreasing.Letusmarkthe rstdk1;k2:=k1+2k2+1vertices(orallofthem,iftherearelessthank1+2k2+1suchvertices).Supposethatthewitnessuofvisnotmarked.Sincej!(X;Y)jk1+2k2,thereisamarkedvertexu02Kn!(X;Y).Bythewaytheverticesareordered,thedistanceofthesubtreeofu0fromxisnotlargerthanthedistanceofthesubtreeofufromx.Therefore,thewitnesspathPconnectingvandugoesthroughtheneighborhoodofu0,i.e.,PhasasubpathP0fromvtoaneighborwofu0.Asu062!(X;Y),theedgewu0isinG0n(X;Y),hencethewitnesspathvP0u0showsthatu0isat-witnessofvwithrespecttoGn(X[u;Y).utThenextlemmaprovesasimilarstatementfort-dangerousvertices.How-ever,nowthemarkingprocedureismorecomplicated.Thereasonforthiscom-plicationisthatat-witnessforvhastosatisfytwo(somewhatcontradicting)requirements:thewitnesshastobereachablefromv(thusithastobeclosetothecliqueQ),butitshouldnotbeaneighborofv(thusitshouldnotbetooclosetoQ).13 Kxb1b2a1b3b4b5a2a3b6a4yu1u4u3u2v1v2v3v4v5v6b 1a 1b 2a 2b 3a 3QFig.5.ProofofLemma14:thepathbetweennodesxandy.Therectanglesshowthesubtreesofthevi'sandui'sonthispath.Lemma14.LetQbeacliqueoft-dangerousvertices.Foreveryk1;k2,thereisaconstantdk1;k2suchthateitherwecan ndanecessarysetorwecanmarkdk1;k2verticesinKsuchthatifXisasetofk1vertices,Yisasetofk2edges,v2Qhasanunmarkedt-witnesswithrespecttoG0n(X;Y),thenvhasamarkedt-witnessu2Kn!(X;Y)aswell.Proof.ConsiderthecliquetreedecompositionofthechordalgraphG0,letTvbethesubtreecorrespondingtoavertexv.SinceQandKarecliques,therearetwonodesxandysuchthateveryv2Qcoversx,andeveryu2Kcoversy.Considertheuniquepathconnectingxandyinthetree,andidentifytheverticesofthepathwiththeintegers1,2,:::,n,wherex=1andy=n.Letu1,u2,:::betheverticesofKhavinglabeltanddenotebyaithesmallestnodeofTuionthispath.Similarly,letv1,v2,:::betheverticesofQanddenotebybithelargestnodeofTvionthispath.Clearly,TviandTujintersectifandonlyifaibj.Forconvenience,weassumethattheai'sandbi'sarealldistinct,thiscanbeachievedbyslightlymodifyingthetreedecomposition.Furthermore,wecanassumethattheverticesareorderedsuchthatthesequenceaiandthesequencebiarestrictlyincreasing(seeFigure5).Wede neasubsequenceofbiandajasfollows.Let 1=1.Foreveryj1,let jbethesmallestvaluesuchthata j�b j.Foreveryi2,let ibethesmallestvaluesuchthatb i�a i 1.Ifwecannot ndsucha ior j,thenwestop.Therefore,thesequenceb 1,a 1,b 2,a 2,:::isstrictlyincreasing.InFigure5,darkrectanglescorrespondtothemembersofthissequence.Letusbeawitnessofat-dangerousvertexv j.Weclaimthatu jisalsoawitnessfort-dangerousvertexv j.Clearly,as�b j(otherwiseuswouldbeaneighborofv j),henceasa jbythede nitionof j.LetPbeawitnesspathfromv jtous.Sinceasa j,pathPgoesthroughtheneighborhoodofu j,i.e.,thereisavertexwofPthatisintheneighborhoodofu j.LetP0bethesubpathofPfromv jtow.Asu jisnotaneighborofv j(byconstruction14 ofthesequenceb 1,a 1,:::),pathv jP0u jisawitnesspath.Thisprovestheclaimthatu jisawitnessofv j.Letb `bethelastelementofthesequencethatcorrespondstoavertexofQ.Weclaimthatif`�2k1+2k2+1,thenwecan ndanecessaryset.LetPibeawitnesspathfromv itoitswitnessu i.Forevery1ik1+k2+1,letHibetheholetv 2iP2iu 2it.Suppose rstthatavertexwofG0appearsintwoholesHiandHi0forii0.ThisisonlypossibleifwisaninternalvertexofbothP2iandP2i0.ItiseasytoseethateachinternalvertexofP2icoversatleastonenodeintheinterval[b 2i;a 2i]andeachinternalvertexofP2i0coversatleastonenodeintheinterval[b 2i0;a 2i0].Therefore,wcoversbotha 2iandb 2i0whichimpliesthatwalsocoversb 2i+1anda 2i+1(since2i0&#x-330;&#x.361;2i+1).Nowtv 2i+1wu 2i+1isaholeofsize4andtheverticesandedgesofthisholeformanecessaryset.Therefore,wecanassumethateveryvertexofG0appearsinatmostoneoftheholesH1,:::,Hk1+k2+1.Thusthereisonlyonevertex,namelyt,thatappearsinmorethanoneoftheholes,hencebyLemma8,(ftg;;)isanecessaryset.Therefore,itcanbeassumedthat`2k1+2k2+1.Foreachi=1;2;:::;`,wemarkthek1+2k2+1verticesu i,u i+1,:::,u i+k1+2k2+1(iftheyexist).Thuswemarkatmostdk1;k2:=(k1+2k1+1)(2k1+2k2+1)vertices.Assumethatvertexvx2Qhasawitnesspath(withrespecttoG0n(X;Y))tosomeuy.Sincevxanduyarenotneighbors,bxayandthereisajwithbxa jay.Ify j+k1+2k2+1,thenuyismarked.Otherwise!(X;Y)doesnotcontainatleastoneoftheverticesu j+1,u j+2,:::,u j+k1+2k2+1,sayvertexu j+r62!(X;Y).Sinceuyisawitnessofvx,thereisapathPfromvxtouyinGn(X;Y)suchthattheinternalverticesofPdonothavelabelt.Froma j+ra j+k1+2k2+1ayitfollowsthatPgoesthroughaneighborwofa j+r;letP0bethesubpathofPfromvxtow.Sinceu j+r62!(X;Y),edgewu j+risinGn(X;Y).Moreover,bxa ja j+rimpliesthatvxandu j+rarenotneighbors,thusvertexu j+risat-witnessofvxwithwitnesspathvxP0u j+r.utInthenexttwolemmas,weextendLemma13andLemma14toapplynotonlyforacliqueQoft-dangerousvertices,butforeverydangerousvertex.ByLemmas11and12,therearenolargeindependentsetsofdangerousvertices.ObservingthatG0ischordalandhenceitscomplementisaperfectgraph(asdiscussedinSection2),weobtainthatthenumberofcliquesrequiredtocoverthedangerousverticesisaconstantdependingonlyonk1;k2.Lemma15.Foreveryk1;k2,thereisaconstantc(1)k1;k2suchthateitherwecan ndanecessarysetorwecanmarkc(1)k1;k2verticesinKsuchthatforeverysetXofk1vertices,setYofk2edges,andlabelt2W,ifvertexvisat-dangerousvertexvwithrespecttoG0n(X;Y)andvhasanunmarkedwitnessu2K,thenvhasamarkedwitnessu02Kn!(X;Y)withrespecttoG0n(X[u;Y).Proof.Foreveryt2W,wemarkverticesasfollows.ConsiderthesetofverticesDthataret-dangerousforKinG0.Forchordalgraphs,amaximumindependent15 setcanbefoundinpolynomialtime[11];letIbeamaximumindependentsetinD.IfjIj�6(k1+k2)2,thenwecan ndanecessarysetbyLemma11.ThusthesizeofthemaximumindependentsetinDisatmostaconstantdependingonlyonk1andk2.ThenumberofcliquesrequiredtocoverDisexactlythenumberofindependentsetsrequiredtocoverDinthecomplementgraph,i.e.,itisthechromaticnumberofthecomplementofG[D].SinceG[D]inducesachordalgraph(asDVnW)andthecomplementofachordalgraphisaperfectgraph[11],itfollowsthatDcanbecoveredbyatmost6(k1+k2)2cliques.ForeachsuchcliqueQ,wemarktheverticesgivenbyLemma13.HencethetotalnumberofmarkedverticesinKcanbeboundedbyaconstantdependingonlyonk1;k2.utLemma16.Foreveryk1;k2,thereisaconstantc(2)k1;k2suchthateitherwecan ndanecessarysetorwecanmarkc(2)k1;k2verticesinKsuchthatforeverysetXofk1vertices,setYofk2vertices,andlabelt2W,ifavertexvist-dangerouswithrespecttoGn(X;Y)andhasanunmarkedwitnessu2K,thenvhasamarkedwitnessu2Kn!(X;Y)withrespecttoG0n(X[u;Y)aswell.Proof.TheproofissimilartotheproofofLemma15.Foreacht2WandeachcliqueQoft-dangerousvertices,wemarkverticesasinLemma14,therestoftheproofisidentical.ut5.4FragmentsofaholeLetHbeaholeinG.SinceGnWischordal,HhastocontainatleastonevertexofW.HenceHnWisasetofpathsP1,P2,:::,Ps,thesetF=H\WtogetherwiththiscollectionofpathswillbecalledthefragmentsoftheholeH(Figure6).ThepathsP1,:::,Psareindependent:PiandPjdonothaveadjacentverticesifi=j.TheinternalverticesofapathPidonothaveanylabelsfromF.Moreover,eachendpointhasexactlyonelabelfromF.TheonlyexceptionisthatifapathPiconsistsofonlyasinglevertex,inthiscaseitcontainsexactlytwolabelsfromF(seeP1inFigure6).AlabelinFcanappearonlyonatmosttwoverticesinthefragments:ifavertexofWisinthehole,thenatmosttwoofitsneighborscanbelongtothehole.However,theneighborsofavertexinWcanalsobeinW,thusitispossiblethatalabelinFappearsononlyoneoronnoneofthepaths.AnotherpropertyisthatifthelengthofPiis1,thenthelabelsofthetwoendpointsaredi erent,otherwisetheholewouldinduceatriangle.Thefollowinglemmashowsthatifwehavethefragmentsofahole,andapathisreplacedwithsomenewpathsatisfyingcertainrequirements,thenthenewcollectionofpathsalsoinducesahole.Lemma17.LetF,P1,:::,PsbethefragmentsofaholeH.AssumethatthelengthofP1isatleast1.LetxandybetheendpointsofP1,andlet`xand`ybetheir(unique)labelsinF,respectively.LetP01beapathwiththefollowingproperties:16 WFP3P2P1Fig.6.ThefragmentsF,P1,P2,P3ofahole.{theendpointsofP01arexandy0,forsomevertexy0thathaslabel`y,{theinternalverticesofP01donothavelabel`x,{if`x=`y,theny0doesnothavelabel`x,{if`x=`y,thenxandy0arenotneighbors.ThenthereisaholeinthegraphinducedbytheverticesofF,P01,P2,:::,Ps.Proof.Weconsidertwocases.IfjFj=1,then`x=`y.Sincexandy0arenotneighbors,theinternalverticesofthepathP01donothavelabel`x,itfollowsthatthepathP01andtheonlyvertexofFformaholeoflengthatleast4.NowassumethatjFj�1.ItcanbeassumedthatP01isaminimalpath,i.e.,eachinternalvertexonthepathisadjacentonlytothepreviousandthenextvertex.Letzbethe(unique)neighborofxonP01.ThepathsP01,P2,:::,Ps,andthesetFgivesawalkfromzto`xwithoutgoingthroughx.Furthermore,zand`xaretheonlyverticesonthiswalkthatareintheneighborhoodofx.Toseethis,observethatxisadjacentonlyto`xinF,onlytozinP01,andtonovertexinP2,:::,Ps.As`xandzarenotadjacent(zdoesnothavelabel`x),Prop.3impliesthatthegraphinducedbyF,P01,P2,:::,Pscontainsahole.utToshowthatavertexu2Kisirrelevant,wehavetoshowthateverysize-(k1;k2)holecoverofGnuisaholecoverofG.Thatis,ifXisasetofk1vertices,Yisasetofk2edges,andthereisaholeHinGn(X;Y)goingthroughu,thenthereisaholeH0inGn(X[u;Y).TheideaistolookatthefragmentsofHandrerouteoneofthepaths:ifpathP1isgoingthroughu,thenwe ndapathP01avoidingu,anduseLemma17toobtaintheholeH0.AsweshallseeinLemma19,ifthelengthofP1isatleast1,thenP01canbefoundusingourpreviousresultsondangerousvertices.However,wehavetotreatseparatelythecasewhenP1consistsofonlyasinglevertex.Thisseeminglysimplecaseturnsouttobesurprisinglydicult.17 Lemma18.Foreveryk1;k2,thereisaconstantc(3)k1;k2suchthateitherwecan ndanecessarysetorwecanmarkc(3)k1;k2verticesinKsuchthatifXisasetofk1vertices,Yisasetofk2edges,andthereisaholeinGn(X;Y)withfragmentsF,P1,:::,PswhereP1isonlyasinglevertexu2K,thenGn(X;Y)hasaholethatdoesnotuseanyunmarkedvertexofK.Proof.Forevery`1;`2;`32W,considerthoseverticesofKthathavebothlabels`1and`2,butdonothavelabel`3andletusmarkk1+2k2+1ofthesevertices(iftherearelessthank1+2k2+1suchvertices,thenwemarkallofthem).Sincethenumberoftriples(`1;`2;`3)dependsonlyonjWjk1+2k2+1,thenumberofmarkedverticescanbeboundedbyafunctionofk1;k2.LetF,P1,:::,PsbethefragmentsofaholeH.Withoutlossofgenerality,assumethatP1consistsofasinglevertexu,inthiscaseuhastwolabels`1,`2fromF.LetusconsiderthecasejFj�2 rst.IfjFj�2,thenthereisanotherlabel`32Fnf`1;`2g.Vertex`3hastwoneighborsaandbintheholeH,andthereisawalkfromatobsuchthattheinternalverticesofthiswalkarenotneighborsof`3.Bythewaywemarkedthevertices,thereisamarkedvertexu02Kn(X;Y)thathaslabels`1;`2,butdoesnothavelabel`3.Therefore,ifwereplaceP1withthepathP01consistingonlyofthesinglevertexu0,thenwegetanotherwalkfromatob.Sinceu0doesnothavelabel`3,itremainstruethattheinternalverticesofthiswalkarenotneighborsof`3.HencebyProp.3,thereisawalkthatcontainsonlythemarkedvertexu0fromK.ThehardcaseiswhenjFj=2,therestoftheproofisdevotedtohandlethissituation.Wemarksomeadditionalverticesasfollows.IfjFj=2,thenscannotbelargerthan2.Furthermore,itisnotpossiblethats=1,sincethatwouldimplythattheholehasonlythreevertices`1;`22F,andP1.Therefore,(*)holeHhastwofragmentsP1andP2,whereP1isonlyasinglevertexofK.ConsideracliquetreedecompositionofG0andletxbeanodethatiscoveredbyeveryvertexofthecliqueK.Assumethatxistherootofthetreeinthedecomposition.ForeachholeHsatisfying(*),de newHtobethenodethatiscoveredbysomevertexofP2andisclosesttothenodex.ObservethatwHcannotbex:thatwouldimplythatsomevertexofP2isadjacentwitheveryvertexofK,includingP1.Letw1,:::,wrbethosenodesthatcanarisethiswayfromsomeholesatisfying(*).Althoughthenumberofholessatisfying(*)canbeexponential,foreverynodewwecancheckinpolynomialtimewhetherthereisaholeHwithwH=w:allwehavetodoistotryeverypossiblesingle-vertexpathP1inKandeverypossibleendpointsofP2,andforeachpossibilitycheckwhetherthereisasuitablepaththatcoversonlywandsomeofitsdescendants.Assumethatthenodeswiareorderedbynonincreasingdistancefromx.Weselectasubsetofthesenodesthefollowingway:wegothroughthelistw1,:::,wr,andaselectanodeifandonlyifnoneofitsdescendantsareselected.Letwi1,:::,wiqbetheselectednodes.Observethataselectednodecannotbetheancestorordescendantofsomeotherselectednode.18 Weconsidertwocases.Firstweshowthatifq�k1+k2,thenanecessarysetcanbeidenti ed.ConsidertheholesHi1,:::,Hik1+k2+1thatgiverisetothenodeswi1,:::,wik1+k2+1.ForeachholeHij,thereisapathP2inthefragmentsofthehole,denotebyPijthispath.Bythede nitionofwij,theverticesofPijcoveronlythedescendantsofwij,henceinparticulartheydonotcoveradescendantofwij0foranyj=j0.Itfollowsthatthereareatmostk1+k2+3verticesthatappearinmorethanoneoftheseholes:thevertices`1;`2andatmostk1+k2+1verticesinK.ThusbyLemma8,wecan ndanecessaryset.Assumethereforethatqk1+k2.Foreachwij,wemarkatmostk1+2k2+1verticesofK.ConsiderthoseverticesofKthathavebothlabels`1and`2.Foreverysuchvertexv,thetreecorrespondingtovhassomedistancefromnodewij.Ordertheverticessuchthatthisdistanceisnonincreasingandmarkthe rstk1+2k2+1verticesinthisordering(orallofthem,iftherearelessthank1+2k2+1suchvertices).Thusatmost(k1+k2)(k1+2k2+1)verticesaremarked.Weshowthatthemarkedverticessatisfytherequirements.LetHbeaholeinGn(X;Y)andletP1,P2bethetwofragmentsofH,whereP1consistsofasinglevertexu2K.SinceHsatis es(*),thereisanodewicorrespondingtoH.Becauseofthewaythenodesareselected,somedescendantofwi(possiblywiitself)isselected,i.e.,somewijisthedescendantofwi.VertexuisnotadjacenttoanyvertexofP2,henceudoesnotcoverwi,i.e.,thetreeofuhasnonzerodistancefromwi.Thismeansthatthetreeofuhasnonzerodistancealsofromwij.Considerthek1+2k2+1verticesmarkedwhenthenodewijwasconsidered.Ifuisnotmarked,thenthismeansthattherearek1+2k2+1verticesinKwhosetreeshavenotsmallerdistancefromwij,implyingthattheseverticesdonotcoverwieither.Atleastoneofthesek1+2k2+1verticesarenotin!(X;Y),letu02Kbesuchavertex.Nowu0isnotadjacenttoanyvertexofP2,hencewecanobtainaholeavoidinguinGn(X;Y)byreplacingP1withthesingle-vertexpathconsistingofu0only.utNowwearereadytoprovethemainlemma:Lemma19.Foreveryk1;k2,thereisaconstantck1;k2suchthateitherwecan ndanecessarysetorwecan ndanirrelevantvertexineverymaximalcliqueofsizegreaterthanck1;k2.Proof.GivenamaximalcliqueK,wemarktheverticesaccordingtoLem-mas15,16,and18.Moreover,foreach`1;`22F,considerthoseverticesthathavelabel`1,butdonothavelabel`2,andmarkk1+2k2+1ofthesevertices(iftherearelessthank1+2k2+1suchverticesforagiven`1;`2,thenallofthemaremarked).Wearguethatanyunmarkedvertexisirrelevant.Sincethenumberofmarkedverticesdependsonlyonk1;k2,thelemmafollows.Letu2Kbeanunmarkedvertex.Toshowthatuisirrelevant,assumethatXisasetofk1vertices,Yisasetofk2edges,andHisaholeinGn(X;Y)containingu.WehavetoshowthatGn(X;Y)containsaholeavoidingu.WeconstructtheholeavoidingubyreplacingthefragmentofHgoingthroughuwithsomeotherpathgoingthroughK.19 Case5 x=y=u Case1`1`1`2`2 Case2`x=`yCase6 yyuuCase3Case4y=uy=uy=uxxxxx`x`x`x`x`y`y`y`yFig.7.ThecasesintheproofofLemma19.LetF,P1,:::,PsbethefragmentsofH.Sincethepathsofthefragmentsareindependent(i.e.,theverticesontwodi erentpathsarenotneighbors),withoutlossofgeneralityitcanbeassumedthatuisinP1andonlyP1intersectsthecliqueK.LetxandybethetwoendverticesofP1.PathP1cancontainatmostoneothervertexofKbesidesu.Weconsiderseveralcasesdependingonwhichcombinationofx=y,u=x,u=y,jK\P1j=1holds(Figure7):Case1:P1consistsofonlyasinglevertex(x=y=u).Lemma18ensuresthatthereisaholeinGn(X;Y)thatdoesnotuseu.Intheremainingcasesweassumethatx=y.Moreover,withoutlossofgeneralityitcanbeassumedthatu=x.Let`xbethe(unique)labelofxinFandlet`ybethe(unique)labelofyinF.Case2:P1consistsoftwoverticesx,y=u,andP1iscompletelycontainedinK.Inthiscase`x=`y,otherwisetherewouldbeatriangleinthehole.Sinceuisnotmarked,therearek1+2k2+1markedverticesinKthathavelabel`ybutdonothavelabel`x.Atleastoneoftheseverticesarenotin!(X;Y),letu0besuchavertex.IfwereplaceP1=fx;ugwiththepathP01=fx;u0g,thenbyLemma17thereisaholenotcontainingu.IntheremainingcasesweassumewithoutlossofgeneralitythatendpointxisnotinK.Case3:x;y62K.Inthiscase,jK\P1jcanbeeither1or2(Fig.7sketchesjK\P1j=2).Itispossiblethat`x=`yandthefollowingproofworksforthat20 situationaswell.Vertexx(resp.,y)isan`x-dangerous(resp.,`y-dangerous)vertexwithrespecttoG0n(X;Y)forK,anduisawitnessforthat.Bythewaytheverticesaremarked(seeLemma15)thereisamarkedwitnessux(resp.,uy)inKn!(X;Y)forx(resp.,y);letPx(resp.,Py)bethecorrespondingwitnesspathinG0n(X[u;Y).Weconsiderthreecases:{Pxnxcontainsavertexy0thathaslabel`y.(NoticethatPxnxcontainsnovertexwithlabel`x,hencethiscaseisnotpossibleif`x=`y).Lety0bethe rstvertexonPx(startingfromx)withlabel`y.LetP01bethesubpathofPxfromxtoy0.NowF,P01,P2,:::,PssatisfytherequirementsofLemma17,henceGn(X;Y)hasaholedisjointfromu.{ThecasewhenPynycontainsavertexthathaslabel`xfollowsbysymmetry.{AssumethatPxnxcontainsnovertexwithlabel`yandPynycontainsnovertexwithlabel`x.LetP01bethepathxPxuxuyPyy;fromux;uy2Kn!(X;Y)itfollowsthatedgeux;uy62Y,henceP01isfullycontainedinGn(X[u;Y).ItiseasytoseethatF,P01,P2,:::,PssatisfytherequirementsofLemma17,henceGn(X;Y)hasaholedisjointfromu.Intheremainingcases,weassumethatx62Kandy2K.Case4:x62K,y2K,u=y(hencejK\P1j=2).Vertexxisan`x-dangerousvertexforK,anduisawitnessforxinG0n(X;Y).Bythewaytheverticesaremarked(seeLemma15)thereisanotherwitnessu02Kn!(X;Y);letPxbethewitnesspathcorrespondingtou0.LetP01bethepathxPxu0y,sinceu02Kn!(X;Y),theedgeu0yisinG0n(X;Y).NowF,P01,P2,:::,PssatisfyLemma17,thusthereisaholenotcontainingu.Case5:x62K,y=u,`x=`y.Inthiscase,jK\P1jcanbeeither1or2(Fig.7sketchesjK\P1j=1).Vertexxisan`x-dangerousvertexforK,anduisawitnessforxinG0n(X;Y).Bythewaytheverticesaremarked(seeLemma15)thereisanotherwitnessu02Kn(X;Y);letPxbethewitnesspathcorrespondingtou0.Sinceuisnotmarked,therearek1+2k2+1markedverticesinKthathavelabel`ybutdonothavelabel`x.Atleastoneoftheseverticesarenotin!(X;Y),lety0besuchavertex.LetP01bethepathxPxu0y0.NowtheconditionsinLemma17aresatis ed,hencethereisaholenotcontainingu.Case6:x62K,y=u,`x=`y.Inthiscase,jK\P1jcanbeeither1or2(Fig.7sketchesjK\P1j=2).Vertexxisan`x-dangerousvertexforK,anduisawitnessforxinG0n(X;Y).Bythewaytheverticesaremarked(seeLemma15)thereisanotherwitnessu02Kn!(X;Y);letPxbethewitnesspathcorrespondingtou0.ItisclearthatF;P01satisfyLemma17.ut6ConclusionsWehaveshownthatChordalDeletionis xed-parametertractable.Theproblemwasformulatedinawaythatincludesboththevertexandedgedeletionversions:k1verticesandk2edgeshavetobedeletedtomakethegraphchordal.Thisformulationcouldbeconvenientforthestudyofotherdeletionproblemsaswell.Ouralgorithmdoesnotprovideaproblemkernelinanobviousway,thus21 itisanaturalopenquestionwhetherthereisproblemkernelofpolynomialsizefortheproblem.Theparameterizedcomplexityliteraturecontainsagrowingnumberof xed-parametertractabilityresultsforvariousdeletionproblems.SomeoftheseresultsfollowimmediatelyfromthegraphminorstheoryofRobertsonandSeymour(see[1]),whilesomeoftheresultsaremoreconcretealgorithms[6,24,22].Recently,ahardnessresulthasbeenobtained,whichshowsthatwecannotexpectthatthedeletionproblemisFPTforeverynaturalgraphclass:Lokshtanovhasshownthatdeletingkedges/verticestomakethegraphwheel-freeisW[2]-hard[20].Thus,despitethesimilarityofwheel-freeandchordal(i.e.,hole-free)graphs,thedeletionproblemisW[2]-hardfortheformerandFPTforthelatter.Anaturalnextstepwouldbetostudythedeletionproblemforintervalgraphs.The(edge)completionproblemforintervalgraphswasshowntobeFPTbyHeggernesetal.[14].Thealgorithmismuchmoreinvolvedthanchordalcompletion.First,alltheminimalchordalcompletionsareenumerated(usingthealgorithmdiscussedintheintroduction),thustheproblemisreducedtochordalgraphsthatarenotintervalgraphs.Thealgorithmisbasedonathoroughunderstandingofsuchgraphs.Itisnotclearwhetherasimilarstrategycouldbeusedfortheintervaldeletionproblem:thealgorithmpresentedinthispapercannotbemodi 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