Properties of Chordal Graphs文档格式.docx
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Properties of Chordal Graphs文档格式.docx
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A/PTayTiongSeng
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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.
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
Assumethatthereexistsanintervalthatisnottriangulated.Thisimpliesthattherewecancreateacycleoflengthgreaterthan3whichdoesnotcontainachord.
ThereareonlyafewwaystoconstructanintervalrepresentationofaP3.Letthe3verticesbea,bandc,withbbeingthevertexthatisconnectedtobothaandc.
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
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