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Characterizing Graph Classes by Intersections of Neighborhoods

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Characterizing Graph Classes by Intersections of Neighborhoods Volume 7, Number 1, Pages 15{20 ISSN 1715-0868 CHARACTERIZING GRAPH CLASSES BY INTERSECTIONS OF NEIGHBORHOODS TERRY A. MCKEE Abstract. The interplay between maxcliques (maximal cliques) and intersections of closed neighborhoods leads to new types of characteri- zations of several standard graph classes. For instance, being hereditary clique-Helly is equivalent to every nontrivial maxclique Q containing the intersection of closed neighborhoods of two vertices of Q, and also to, in all induced subgraphs, every nontrivial maxclique containing a sim- plicial edge (an edge in a unique maxclique). Similarly, being trivially perfect is equivalent to every maxclique Q containing the closed neigh- borhood of a vertex of Q, and also to, in all induced subgraphs, every maxclique containing a simplicial vertex. Maxcliques can be general- ized to maximal cographs, yielding a new characterization of ptolemaic graphs. 1. Maximal cliques and closed neighborhoods A clique of a graph G is a complete subgraph of G or, interchangeably, the vertex set of a complete subgraph. A clique Q is a p-clique if jQj = p; is a maxclique if Q is not contained in a (jQj + 1)-clique; and is a simplicial clique if Q is contained in a unique maxclique. Simplicial 1-cliques are traditionally called simplicial vertices, and so simplicial 2-cliques can be called simplicial edges. The open neighborhood NG(v) of a vertex v is the set fw 2 V (G): wv 2 E(G)g. The closed neighborhood NG[v] of v is the set NG(v) [ fvg or, interchangeably, the subgraph of G induced by the set NG[v]. Lemma 1. For every graph G and every p ≥ 1, a maxclique Q of G contains NG[v1] \···\ NG[vp] for distinct v1; : : : ; vp 2 Q if and only if Q contains a simplicial p-clique of G. For k ≥ 2, define a k-ocular graph to consist of 2k vertices that are partitioned into W = fw1; : : : ; wkg and U = fu1; : : : ; ukg where W induces a k-clique and each NG(ui) = fwj : j 6= ig (the subgraph induced by U can contain any subset of edges uiuj); see Figure 1. This is the terminology of Received by the editors June 24, 2010, and in revised form May 25, 2011. 2000 Mathematics Subject Classification. 05C75, 05C69. Key words and phrases. Maximal clique, closed neighborhood, clique-Helly graph, triv- ially perfect graph, simplicial vertex, simplicial edge, ptolemaic graph. c 2012 University of Calgary 15 16 TERRY A. MCKEE ∼ [2] when k ≥ 3. The only 2-ocular graphs are the path u1; w2; w1; u2 = P4 ∼ and the cycle u1; w2; w1; u2; u1 = C4. 2 TERRY A. MCKEE u u 3aa !! 4 L@ aa! u u u2 w3 u1 w ! w 2 1 L 1 2 @ @ w1 w2 L @ L LL @ L @ w1 w2 w4 !w3 L u3 !aa !! aa@LL u2 u1 Figure 1. The smallest of the two 2-ocular graphs, graphs, of of the the four 3-ocular graphs, and of the the eleven eleven 4-ocular 4-ocular graphs. graphs. For p ≥ 2, define a graph G to be pp-clique-Helly-clique-Helly when,when, for for every every family family F of maxcliques≥ of G, if every p members of F have an an element element in in common, common, thenF all the members of F have an elementF in common. If If every every induced induced subgraph of G is p-clique-Helly,F then G isis called called hereditary p p-clique-Helly-clique-Helly.. Reference [2] contains several characterizations of hereditary hereditary pp-clique-Helly-clique-Helly graphs, including condition (2.4)(4) in in Theorem Theorem 2. 2. Theorem 2. The following are equivalent for every graph graph G and p ≥ 22:: 0 ≥ (1)(2.1):If QIfisQ ais maxclique a maxclique of anof an induced induced subgraph subgraphGG0ofofGGandand jQQj ≥ pp,, 0 0 | | ≥ then Q contains NNGG [[vv11]]\···\ NG [[vp]] forfor distinct distinct v1;, :. :. :. ;, vp 2 Q. 0 ∩ · · · ∩ 0 0 ∈ (2)(2.2):If QIfisQ ais maxclique a maxclique of anof an induced induced subgraph subgraphGG0ofofGGandand jQQj ≥ pp,, 0 | | ≥ then Q contains a simplicial pp-clique-clique of of GG.0. (3)(2.3):G isG hereditaryis hereditaryp-clique-Helly.p-clique-Helly. (4)(2.4):G containsG contains no induced no induced(p +( 1)p-ocular+ 1)-ocular subgraph. subgraph. Proof. SupposeSupposepp≥ 2. Lemma Lemma 1 1 implies implies the the equivalence equivalence (2 (1).1) ,(2(2)..2). The The equivalence (2(3).3), (4)≥(2.4) is [2, is [2, Thm. Thm. 4]. 4]. ⇔ ⇔ 0 To prove (4)(2.4)) (1),(2.1), suppose suppose (4) (2.4) holds, holds,G isG an0 is induced an induced subgraph subgraph of G, ⇒ 0 andof GQ, andis a maxcliqueQ is a maxclique of G with of jGQ0j >with p (theQ jQ>j p=(thep caseQ being= p immediate).case being 0 | | | | Letimmediate).Q = fw Let1; : :Q : ;0 w=p+1wg1 ⊆, . .Q . , wand,p+1 wheneverQ and, 1 whenever≤ i ≤ p 1+ 1,i letpS+i 1,= T 0 NG0 [wj]. Thus Q ⊆{ Si for each i}. If ⊆ for each i there exists≤ a vertex≤ ui 2 letj6=Sii = j=i NG0 [wj]. Thus Q0 Si for each i. If for each i there exists 6 0 ⊆ Sai vertex− Q, thenu WS = QQ,and thenUW= f=u1Q; : :and : ; upU+1=g wouldu , . induce . , u a (wouldp + 1)-ocular induce Ti ∈ i − 0 { 1 p+1} subgrapha (p + 1)-ocular (noting subgraph that ui and (notingwi are that notui adjacent,and wi are since notQ adjacent,is a maxclique). since Q Thereforeis a maxclique). (4) implies Therefore that some (2.4) impliesSi = Q that; without some lossSi = ofQ generality,; without loss say Tp p Sp+1 = Q. That makes Q = NG0 [wj] with distinct w1; : : : ; wp 2 Q, of generality, say Sp+1 = Q. Thati=1 makes Q = i=1 NG0 [wj] with distinct showingw1, . , w thatp Q (1), showing holds. that (2.1) holds. To prove prove∈ (2 (1).1)) (2(4),.4), suppose suppose (4)(2.4) fails fails because becauseT GGcontainscontains an induced induced (p + 1)-ocular subgraph⇒ G0 where V (G0) is partitioned into into WW [UU asas in in the the 0 0 ∪ definition of (p + 1)-ocular. Each NG0[[wi]] contains contains the the pp verticesvertices uuj thatthat T G0 i j have j 6= i. Therefore, each uj 2 NG0 [wi], and so Q = W can never have j = i. Therefore, each uj ii=6 jj NG0 [wi], and so Q = W can never 6 ∈ 6 contain the intersection of p neighborhoods NG0[[wj]] with with wwj 2 QQ,, showing showing T G0 j j ∈ that (1)(2.1) fails. fails. 2 Hereditary 2-clique-Helly graphs are called hereditary clique-Helly graphs in [10] and clique reducible graphs in [11]. These papers contain several CHARACTERIZING GRAPH CLASSES 17 Hereditary 2-clique-Helly graphs are called hereditary clique-Helly graphs in [10] and clique reducible graphs in [11]. These papers contain several other characterizations, and [8] will contain more characterizations involving simplicial cliques. Corollary 3. The following are equivalent for every graph G: (1) If Q is a maxclique of an induced subgraph G0 of G and jQj ≥ 2, 0 0 then Q contains NG0 [v] \ NG0 [v ] for distinct v; v 2 Q. (2) Every nontrivial maxclique of an induced subgraph of G contains a simplicial edge. (3) G is hereditary clique-Helly. (4) G contains no induced 3-ocular subgraph. Proof. This follows from the p = 2 case of Theorem 2. (The equivalence (3) , (4) also appears in [10, 11].) A graph G is called trivially perfect if, for every induced subgraph G0 of G, the cardinality of the largest independent set in G0 equals the num- ber of maxcliques in G0. Reference [4] contains several other characteriza- tions, including condition (4) in Theorem 4. See [1, 9] for many additional characterizations|and names|for trivially perfect graphs; [8] will contain more characterizations involving simplicial cliques. Theorem 4. The following are equivalent for every graph G: (1) If Q is a maxclique of an induced subgraph G0 of G, then Q contains NG0 [v] for some v 2 Q. (2) Every maxclique of an induced subgraph of G contains a simplicial vertex. (3) G is trivially perfect. (4) G contains no induced P4 or C4 subgraph. Proof. The equivalence (1) , (2) follows from Lemma 1. The equivalence (3) , (4) is [4, Thm. 2]. The equivalence (1) , (4) can be proved by a simple modification of the proof of (1) , (4) in Theorem 2, taking p = 1 and using that P4 and C4 are the two 2-ocular graphs. Corollary 5 will be a restriction of Corollary 3 to chordal graphs|the graphs in which every cycle of length four or more has a chord (see [1, 9] for other many characterizations and history). Note that the existence of an edge uiuj in a p-ocular graph (with vertex set partitioned into W and U as in the definition) would produce a chordless 4-cycle ui; uj; wi; wj; ui. Therefore a p-ocular graph is chordal if and only if the set U is independent (meaning that there are no edges between vertices in U). A disk DG[v; k] of a graph G is the set fx : 0 ≤ d(v; x) ≤ kg ⊆ V (G), where d(v; x) denotes the v-to-x distance in G. Define G to be disk-Helly when, for every family F of disks of G, if every two members of F have an element in common, then all the members of F have an element in common.
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