October 2003 These notes present some properties of chordal graphs a set of undirected graphs that are important for undirected graphical models De64257nitions We consider undirected graphs VE where is the vertex set and the edge set is a set of ID: 74383
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Avertexissimplicialinagraphifitsneighborsformacompletesubgraph.Agraphisrecursivelysimplicialifitcontainsasimplicialvertexvandwhenvisremovedthesubgraphthatremainsisrecursivelysimplicial.EquivalenceTheoremTheorem1.ThefollowingpropertiesofGareequivalent.1.Gischordal.2.Gisdecomposable.3.Gisrecursivelysimplicial.4.Ghasajunctiontree.Weshallpresenttheproofasaseriesoflemmas(1=)2=)3=)4=)1).Lemma1.GischordalimpliesGisdecomposable.Proof.Weprovebyinductionthateverychordalgraphwithnverticesisde-composable.Thisistriviallytrueforn=1.Ifitistrueforanyn,thenthefollowingargumentshowsthatitistrueforagraphGwithn+1vertices.Step1:IfGiscomplete,itisdecomposable.SosupposethatGisnotcomplete.Step2:WecanexpressVasthedisjointunionV=A[B[S,whereSseparatesAfromBinGandA;Barenonempty.Indeed,sinceGisnotcomplete,Vcontainsa;bthatarenotneighbors.LetSVbeaminimalsetthatseparatesafromb.(NoticethatSmightbeempty).LetAbethesubsetofV�SconnectedtoabysomepathinV�S,andletBbetheremainder,B=V�S�A.Clearly,SseparatesAfromBinG.Step3:Siscomplete.3a:WemayassumethatShascardinalityatleast2,forotherwiseitistriviallycomplete.3b:Foranytwodistinctnodesu;vinS,therearepathsu;a1;:::;an;vandu;b1;:::;bm;vwithai2A,bi2Bandn;m1.Indeed,sinceSwasaminimalsetthatseparatesafromb,theremustbeapathfromatouandfromatov,sincetheabsenceofoneofthesepathswouldimplythatSwasnotminimal.3c:uandvareneighbors.Toseethis,takethepathfromutovthroughAthathasminimallength,andsimilarlythepathfromutovthroughBofminimallength.Thispairofpathsformsacycle,whichmusthaveachord.Toseethatthechordmustbebetweenuandv,noticethattheminimalityofthepathsimpliesthatthechordcannotbebetweenverticesthatarebothinA,norcanitbebetweenverticesthatarebothinB.Inaddition,thefactthatSseparatesAfromBimpliesthechordcannotbebetweenavertexinAandoneinB.Step4:ThesubgraphsinducedbyA[SandB[Sarechordal.Indeed,ifoneofthesesubgraphscontainsachordlesscycle,thensodoesG.2 Assumingthatthestatementistrueforsomevalueofn,consideragraphwithajunctiontreeTcontainingn+1nodes.FixaleafCofT,andletC0betheneighborofCinT,andletT0bethetreethatremainswhenCisremoved.Step1:IfCC0,thenT0isajunctiontreeforG.Step2:Ontheotherhand,ifC\C0C,removingthenonemptysetR=C�C0fromVleavesasubgraphG0thatischordal.Toseethis,noticethatRhasanemptyintersectionwitheverycliqueinT0.ItiseasytoseethatT0isajunctiontreeforG0,andsoG0ischordal.Step3:ItfollowsthatGcontainsnochordlesscycles.Indeed,ifacycleisentirelyinG0,itisnotchordless.IfthecycleisentirelyinthecompletesubgraphdenedbyC,itisnotchordless.IfthecycleintersectsR,C\C0,andV�C,thensincethesubgraphdenedbyC\C0iscomplete,thecyclehasachord. UndirectedgraphicalmodelswithchordalgraphsTheorem2.ThefollowingpropertiesofGareequivalent.1.Gischordal.2.ThereisaneliminationorderingforwhichthegraphGisaxedpointoftheUndirectedGraphEliminatealgorithm.3.ThereisanorientationoftheedgesofGthatgivesadirectedacyclicgraphwhosemoralgraphisG.4.Thereisadirectedgraphicalmodelwithconditionalindependenciesiden-ticaltothoseimpliedbyG.Wesketchtheproofoftheseimplications.Deneasimplicialvertexsequenceasanorderingofthenodesofarecursivelysimplicialgraphthatexhibitstherecursivelysimplicialityofthegraph.Thatis,asweprogressivelyremovethenodesinthisorder,thenextnodeintheorderissimplicialintheremainingsubgraph.Eliminationordering:Eliminatinganodeleavesthegraphunchangedi thenodeissimplicial.Thus,theexistenceofaneliminationorderingthatleavesthereconstitutedgraphidenticalisequivalenttotheexistenceofasimplicialvertexsequence.DAGwithsamemoralgraph:GivenarecursivelysimplicialgraphG,wecanconstructaDAGGDasfollows.Fixasimplicialvertexsequence(v1;:::;vn)forG.DeneG1=G.Atstept,addvtanditsneighborsinGttoGD(iftheyarenotalreadypresent),andaddthecorrespondingedgestoGDsothattheyaredirectedtowardsvt.ThensetGt+1tothesubgraphofGtthatremainswhenvtisremoved.ItisclearthatGDisacyclic,sincetheedgesaredirectedinaxedorder.Also,byconstruction,themoralgraphrequiresnoadditionaledges.4 ReferencesTheclassicpaperonthegraph-theoreticequivalencespresentedhereisC.Beeri,R.Fagin,D.Maier,andM.Yannakakis,Onthedesirabilityofacyclicdatabaseschemes,JournaloftheACM,30(3):479{513,July1983.SeealsoJ.Pearl,ProbabilisticReasoninginIntelligentSystems,MorganKaufmann,1988.AndChapters16and17ofthetext,M.Jordan,AnIntroductiontoProbabilisticGraphicalModels.6