Properties of Chordal Graphs.docx

Properties of Chordal Graphs.docx

《Properties of Chordal Graphs.docx》由会员分享，可在线阅读，更多相关《Properties of Chordal Graphs.docx（12页珍藏版）》请在冰豆网上搜索。

PropertiesofChordalGraphs

PropertiesofChordalGraphs

UndergraduateResearchOpportunitiesProgrammeinScience

（UROPS）

Semester2,2001/2002

DepartmentofMathematics

NationalUniversityofSingapore

Supervisor:

A/PTayTiongSeng

Doneby:

JaronPowTienMin（U002626M02）

Acknowledgements

VeryspecialthankstoA/PTayTiongSengforagreeinginthefirstplacetoundertakethisUropsprojectonChordalGraphs.Withouthim,IwouldnothavebeenabletodwellasdeeplyasIdidintothetopicofchordalgraphs.

TakingupGraphTheoryinUROPSwasofmyintialhighestpriorityasitismyfavoritetopicofallintertiarymathematics,mostlyalsoduetothefactthatA/PTaywasthelecturerforMA3233,whichwasthedecidingfactorformelikingthetopicallthewayfromthestart,withfurtherreadingsoutofthesyllabusandinvitationstoatalkbythedeanofMathematicsfromHongKongonGraphTheoryhelpingeverybit.

IwouldliketothankA/PTayagainonhispatienceinguidingmethroughthefineraspectsofprovingtheoremsandlemmas,beitintheverybasicsofGraphtheoryorinthelaterdwellingsontheLexBFSalgorithm.

ThelastthanksgoestomypeersintheSpecialProgrammewhohavehelpedmelookthroughpartsofthereportanddiscussingwithmealsothemanyareasinGraphTheory.

Abstract

Graphtheorystartedintheearly1700swhereEulerdiscussedtheproblemofwhetherisitpossibletocrossthe7bridgesofKonigsbergexactlyonce.Ofcourse,thetopicofGraphtheoryevolvedthroughtheyearssuchthatwenowhaverepresentationslikevertices,edgesandcycles（notethatwhenEulersolvedtheKonigsbergproblem,hedidnotatallusetheconceptsofedgesandverticesatall.Alloftheterminologythatweusenowisaresultofmathematiciansgoingdeeperintothetopicandimplementingthetermsthattheyfindusefultostudyofgraphtheory）.

Currently,manymathematiciansandcomputerscientistsaregoingintographtheoryascertainbranchesofitsstudyareimportantintheirrespectivefields.

Whatwehavehopedtoachieveinthispaperistogodeeperintothestudyofaparticularaspectofgraphtheory,andthechoicewaschordalgraphsasitiscurrentlygainingpopularityincomputerscientists.

Chordalgraphsshowmanylinkstoperfectgraphsandintervalgraphs.Inthispaperisashortprooftohowallintervalgraphsaretriangulated,butmoreimportantly,wetouchedonthetopicofmoplexes,whichservetogeneralizeDirac’stheoremsregardingtriangulatedgraphs.

CONTENTS

1.PRELIMINARIES5

2.INTERVALGRAPHS9

3.RELATIONSHIPOFTRIANGULATEDGRAPHS

TOTHEPERFECTELIMINATIONSCHEME11

4.MOPLEXESINTRIANGULATEDGRAPHS14

5.GENERALIZATIONOFDIRAC’STHEOREM

TOANYGRAPH18

6.REFERENCES21

1.PRELIMINARIES

Inthispaper,thenotationsusedwillbeasfollows.

1.1Graphs

G=（V,E）isafiniteundirectedgraphwithvertexsetVandedgesetE,|V|=n,

|E|=m.N（x）denotestheneighborhoodofvertexx（notethatitdoesnotcontainxitself）.IfN（X）istheneighborhoodofXwhereXV,N（X）={UxєXN（x）\X}.

1.2TriangulatedGraphs

AsimpleGraphGistriangulatedifeverycycleoflength>3hasanedgejoining2nonadjacentverticesofthecycle.Theedgeiscalledachord,andtriangulatedgraphsarealsocalledchordalgraphs.

1.3CliquesandSimplicialVertices

AcliqueofGisasetofpairwiseadjacentvertices.

AvertexvofagraphGisasimplicialvertexifftheinducedsubgraphofN（v）,isaclique.

1.4ThePerfectVertexEliminationScheme

AperfectvertexeliminationschemeofagraphGisanordering{v1,v2,v3,...,vn}suchthatfor1≤i≤n-1,viisasimplicialvertexofthesubgraphofGinducedby{vi,vi+1,vi+2,...,vn}.Itisalsocalledaperfectscheme.

（Remarks）

Anyvertexofdegree1istriviallysimplicial.

Foratree,thereexistatleast2endvertices.Sinceendverticesareofdegree1andhencetriviallysimplicial,everytreehasatleast2simplicialvertices.Afterdeletinganendvertex,westillgetatree.Therefore,everytreehasaperfectvertexeliminationschemeofsequence{v1,v2,v3,...,vn},whereviisanendvertexofthesubgraphwhichisatreeinducedby{vi,vi+1,vi+2,…,vn}

1.5Separation

AsubsetSVofaconnectedgraphGiscalledaseparatoriffG（V\S）is

disconnected.ThesetoftheconnectedcomponentsofG（V\S）isdenotedasCC（S）.Siscalledanab-separatoriffaandbliein2differentcomponentsofCC（S）.Siscalledaminimalab-separatoriffSisanab-separatorandnopropersubsetofSisalsoanab-separator.Siscalledaminimalseparatoriffa,bєVsuchthatSisaminimalab-separator.

1.6Triangulation

AtriangulatedgraphH=（V,EUF）iscalledatriangulationofG=（V,E）.

ThetriangulationisminimaliffforanyedgeeofF,H’=（V,（EUF）\{e}）isnottriangulated.Fisthencalledaminimalfill-in.

UniqueChordProperty

AtriangulationHofGisminimaliffforalleєF,eistheuniquechordofsome4-cycleofH.

ChordisuniqueChordsarenotunique

Aminimalfill-inNotaminimalfill-in

CrossingedgeLemma

Noedgeofaminimalfill-inofGcanjoin2connectedcomponentsinCC（S）,whereSisacliqueseparatorofG（acliqueseparatorisaseparatorthatisaclique）.

Proof:

IfCisacliqueseparatorofagraphG,G–Cconsistsofatleast2separatedcutcomponents.Take2verticesaandbfromthe2cutcomponents.Everycyclecontainingaandbmustconsistsofatleast2verticess,tinS.SinceSisaclique,thecycleissplitinto2smallercyclesinGAandGBrespectivelybecausethecyclecontaininga,bissplitinto2bythes-tedge.Thus,totriangulatethegraphG,theindividualcyclesinGAandGBmustbetriangulatedfirst.Hence,aminimalfill-inwillnothaveanedgethatjoins2connectedcomponents.

Minimalseparatorproperty

LetHbeaminimaltriangulationofG.Anyminimalab-separatorofHisalsoaminimalab-separatorofG.

Proof:

ItcanbeeasilydeducedthatanyseparatorofHisalsoaseparatorofG.LetSbeaminimalab-separatorofH（Sisaclique）andG’beobtainedfromGbeinsertingedgestoSsuchthatSbecomesaclique.Thus,HisatriangulationofG’.Bythecrossing-edgelemma,ifanysubsetS’ofSisanab-separatorofG’,thenitisanab-separatorinH,sincenoedgesaddedjoin2connectedcomponents.Thus,Sisaminimalab-separatorinG’.AndsinceS-xisnotanab-separatorinG,itisaminimalab-separatorofG.

2.INTERVALGRAPHS

Definition:

AgraphG=（V,E）isanintervalgraphiffthereexistsanassignmenttoeachvertexxєVofanintervalJ（x）ontherealnumberlinesuchthatx,yєEJ（x）J（y）.

Proposition

AllIntervalGraphsaretriangulated

Proof:

Assumethatthereexistsanintervalthatisnottriangulated.Thisimpliesthattherewecancreateacycleoflengthgreaterthan3whichdoesnotcontainachord.

ThereareonlyafewwaystoconstructanintervalrepresentationofaP3.Letthe3verticesbea,bandc,withbbeingthevertexthatisconnectedtobothaandc.

J（a）J（c）

J（b）

J（a）J（c）

J（b）

J（a）J（c）

J（b）

Withoutlossofgenerality,thesearetheonly3intervalrepresentationsofaP3.

Inthelattertwocases,anyintervalthatoverlapswithJ（c）willalsooverlapwithJ（b）.Thusthevertexitrepresentswillbeadjacenttob.Inthefirstcase,sincethereisapathfromctoa,oneoftheintervalsrepresentingthispathmustoverlapwithJ（b）andhencethereisachordaswell.

Inordertocreateatrue4-cycle,aninterval（oraseriesofthemforachordlesscycleoflengthgreaterthan4）hastobecreatedthatoverlapsJ（a）andJ（c）butnotJ（b）.Fromtherepresentationsabove,weseethatitisnotpossible.Hence,thereisnochordlesscyclethatisanintervalgraph,whichimpliesthatallintervalgraphsaretriangulated.

3.RELATIONSHIPOFTRIANGULATEDGRAPHSTOTHEPERFECTELIMINATIONSCHEME

Theorem3.1IfSisaminimalab-separator,everyvertexxinSmustbeadjacenttosomevertexaofGAandsomevertexbofGB

ForanyxS,sinceS-xisnotaseparator,GAandGBwillbeconnectedinG-{S-x}.Hence,thereexistsana-bpathwhichcontainsx.Therefore,xmustbeadjacenttosomevertexinGAandsomevertexinGB.

Theorem3.2（Dirac’sTheorem）AgraphistriangulatediffeveryminimalvertexseparatorofGisaclique.

Necessity:

LetthegraphGbetriangulatedandSbeaminimalseparatorofG.LetGAandGBbe2distinctcomponentsofG\S.SinceSisaminimalseparator,everyvertexxinSmustbeadjacenttosomevertexofGAandsomevertexofGB.Hence,foranypairx,yinS,thereexistpathsP1:

xa1…aryandP2:

xb1…bsywhereeachaiєV（GA）andeach

biєV（GB）.AssumingalsothatP1andP2arechosentobeoftheshortestlength,xa1…arybs…b1xisacycleoflengthatleast4,andso（asGistriangulated）mustcontainachord.However,asP1andP2arechosentobeoftheshortestlength,thechordmustbexy.Thus,everypairx,yinSareadjacentandSisaclique.

Sufficiency:

WenowhavetoprovethatifeveryminimalseparatorofGisaclique,everycycleoflengthatleast4inGcontainsachord.AssumethateveryminimalseparatorofGisaclique.Letaxby1y2…yrabeacycleCoflength4inG.IfabwerenotachordofC,denotebySaminimalseperatorthatputsaandbindistinctcomponentsofG\S.ThenSmustcontainxandyjforsomej.Byhypothesis,Sisaclique,andhencexyjisanedgeofG,andthereforeachordinC.Thus,Gistriangulated.

Lemma3.3EverytriangulatedgraphGhasasimplicialvertex.Moreover,ifGisnotcomplete,ithas2nonadjacentsimplicialvertice