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A note on near chordal plane triangulations SameeraMSalam,NandiniJWarrier,DaphnaChacko,K.MuraliKrishnan,SudeepK.S. DepartmentofComputerScienceandEngineering,NationalInstituteofTechnologyCalicut,Kerala,India673601 7 1 0 Abstract 2 n Itisshownthataplaneneartriangulatedgraphischordaliffitdoesnotcontainanyinduced a wheelW forn≥5. n J Keywords: Planetriangulatedgraphs,planeneartriangulatedgraphs,Chordalgraphs 0 1 ] O 1. Introduction C h. A planeembeddingofa (planar)graphG is called a triangulationif the boundaryofevery t faceisacycleoflengththree. Aplaneembeddingofasimplemaximalplanargraphisaplane a triangulation[1]. If the abovedefinitionis relaxedto permittheouterface to havea boundary m of length exceeding three, the embedding is called a near triangulation. A graphG is said to [ containachordlesscycleifithasaninducedcycleCt fort ≥ 4. Agraphthatdoesnotcontain 1 anychordlesscycleiscalledachordalgraph[2].Appoloniannetworksarewellknownexamples v intheliteratureformaximalplanarchordalgraphs[3]. 7 Inthisshortnotewederiveasimplestructuralcharacterizationformaximalplanarchordal 4 graphsthatdependsonlyontheneighbourhoodofindividualvertices.Infacttheproofworksfor 4 3 neartriangulationsaswell. AproofispresentedinSection3. Thefollowingsectionestablishes 0 somenotationanddefinitions. . 1 0 2. Preliminaries 7 1 LetG = (V,E)beasimpleundirectedgraph. Adrawingmapseachvertexu ∈ V toapoint : v ε(u) in R2 and each edge (u,v) ∈ E to a path with endpointsε(u) and ε(v). The drawing is a i X planeembeddingifthepointsaredistinct,thepathsaresimpleanddonotcrosseachotherand the incidencesare limited to the endpoints[2]. The well knownJordan Curve Theoremstates r a that: if J isasimpleclosedcurveinR2, thenR2 − J hastwo components(Int(J)and Ext(J)), with J as the boundary of each [4]. Given a plane triangulationG, Int(G) denotes the set of verticesintheinterioroftheclosedcurvedefinedbytheboundaryofthetriangulation. AwheelW ofordern ≥ 4isthegraph K ∪C . ThecycleC iscalledtherimofthe n 1 n−1 n−1 wheel[2].Aseparatorinaconnectedgraphisasetofvertices,theremovalofwhichdisconnects Emailaddresses:[email protected](SameeraMSalam),[email protected](NandiniJWarrier), [email protected](DaphnaChacko),[email protected](K.MuraliKrishnan),[email protected] (SudeepK.S.) PreprintsubmittedtoElsevier January13,2017 the graph. A clique in a graph is a set of pairwise adjacent vertices. A clique separator is a separatorwhichisaclique[5]. Definition2.1(Separatingtriangle). Acliqueseparatorwiththreeverticesiscalledaseparating triangle. Definition2.2(W−graph). AneartriangulationG,G , K iscalledaW−graphifG doesnot 4 containaseparatingtriangle. LetG beanygraphandv ∈ V(G). DefinetheopenneighbourhoodofthevertexvasN(v)= {v′ ∈V :(v′,v)∈E}. Theclosedneighbourhoodofv,N[v]={v}∪N(v). 3. Nonchordalplaneneartriangulatedgraphs WefirstprovethefollowingLemma: Lemma3.1. IfGisaW−graphthen∀u∈Int(G),N[u]inducesawheelW forsomek≥5. k Proof. Let G be a W−graph, u ∈ Int(G). As G is plane triangulated, it is easy to see that |N(u)| ≥ 3. MoreoverasGisaW−graphandG , K ,|N(u)| , 3forotherwiseverticesinN(u) 4 willformaseparatingtriangle. HenceletN(u) = {u ,u ,...u }forsomek ≥ 5andletG′ be 1 2 k−1 thesubgraphofGinducedbyN(u). TocompletetheproofitisenoughtoprovethatG′ induces acycleinG. Weruleoutthefollowingthreecases. Case1 : SupposeG′ is not connected and has componentsG ,G ,...G where 2 ≤ l ≤ k−1. 1 2 l WithoutlossofgeneralitywemayassumethatG containsvertices{u ,u ,...u}andG 1 1 2 i 2 contains vertices {ui+1,...uj} (1 ≤ i < j ≤ k − 1) in the clockwise order. Then as G is triangulated with u ∈ Int(G) and {(u,ui),(u,ui+1)} ⊆ E(G), it must be the case that (ui,ui+1) ∈ E(G), for otherwise u,ui and ui+1 will be part of a face of length exceeding three,contradictingtheassumptionthatuiandui+1areindifferentcomponents. Case2 : SupposeG′ is a path. With out loss of generality we may assume that the vertices in the path are u ,u ,....u orderedfrom left to right. As {(u,u ),(u,u )} ⊆ E(G) and 1 2 k−1 1 k−1 u ∈ Int(G),itfollowsthatGmusthaveaninternalface f containingedgesu,u andu . 1 k−1 Butas(u ,u )<E(G), f cannotbeatriangle,acontradiction. 1 k−1 Case3 : SupposeG′ contains a vertex u with degree at least three. Let x,y and z be any three i neighboursofu inG′. Thenoneamongthetriplets(u,u,x),(u,u,y)and(u,u,z)willbe i i i i aseparatingtriangleinG,acontradiction. ItfollowsthereforethatG′inducesacycleoflengthk−1 Theobservationbelowfollowsdirectlyfromthedefinitionofaseparatingtriangle. Observation3.1. Lettheverticesu,vandwformaseparatingtriangleinaplaneneartriangu- latedgraphG. Thenthetriangle(u,v,w)separatesGintotwocomponents(sayG andG )with 1 2 (u,v,w)astheboundary.Moreover,GischordaliffbothG andG arechordal. 1 2 Thefollowingtheoremshowsthatchordalplaneneartriangulationsarecharacterizedbythe closedneighbourhoodofinternalvertices. 2 Theorem3.1. A planeneartriangulatedgraphis notchordaliffit containsan inducedwheel W forsomek≥5. k Proof. LetG(V,E)be a near triangulatedgraph. If G containsan induced W for some k ≥ 5 k thentherimofW isachordlesscycleandGisnonchordal. k Conversely, suppose G is non chordal. If G is a W−graph, then the claim follows from Lemma 3.1. Otherwise, let △ = (u,v,w) be a separating triangle in G. Then by the Observa- tion3.1,(u,v,w)dividesGintotwocomponentssayG andG andeitherG orG orbothare 1 2 1 2 notchordal.Theclaimfollowsbyinduction. Acknowledgment We would like to thank Prof Ajith Diwan, IIT Bombay for his valuable suggestions and support. References [1] R.Balakrishnan,K.Ranganathan,Atextbookofgraphtheory,SpringerScience&BusinessMedia,2012. [2] D.B.West,etal.,Introductiontographtheory,Vol.2,PrenticehallUpperSaddleRiver,2001. [3] P.S.Kumar,C.E.V.Madhavan,Anewclassofseparatorsandplanarityofchordalgraphs,in: Proceedingsofthe Ninth Conference on Foundations of Software Technology and Theoretical Computer Science, Springer-Verlag, London,UK,UK,1989,pp.30–43. URLhttp://dl.acm.org/citation.cfm?id=646827.707381 [4] W.Fulton,Algebraictopology:afirstcourse,Vol.153,Springer,1995. [5] A.Berry,R.Pogorelcnik,G.Simonet,Anintroductiontocliqueminimalseparatordecomposition,Algorithms3(2) (2010)197–215. 3