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 characterizations 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 simplicial edge (an edge in a unique maxclique). Similarly, being trivially perfect is equivalent to every maxclique Q containing the closed neighborhood of a vertex of Q, and also to, in all induced subgraphs, every maxclique containing a simplicial vertex. Maxcliques can be generalized 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 |Q| = p; is a maxclique if Q is not contained in a (|Q| + 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 {w ∈ V (G): wv ∈ E(G)}. The closed neighborhood NG[v] of v is the set NG(v) ∪ {v} 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 ∈ 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 = {w1, . . . , wk} and U = {u1, . . . , uk} where W induces a k-clique and each NG(ui) = {wj : j 6= i} (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, trivially 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 |QQ| ≥ pp,, 0 0 | | ≥ then Q contains NNGG [[vv11]]∩ · · · ∩ NG [[vp]] forfor distinct distinct v1, . . . , vp ∈ Q. 0 ∩ · · · ∩ 0 0 ∈ (2)(2.2):If QIfisQ ais maxclique a maxclique of anof an induced induced subgraph subgraphGG0ofofGGandand |QQ| ≥ 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 |GQ0| >with p (theQ |Q>| p=(thep caseQ being= p immediate).case being 0 | | | | Letimmediate).Q = {w Let1, . .Q . ,0 w=p+1w}1 ⊆, . .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 ∈ 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= {=u1Q, . .and . , upU+1}=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 ∈ 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 ∈ 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 ∈ 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 |Q| ≥ 2, 0 0 then Q contains NG0 [v] ∩ NG0 [v ] for distinct v, v ∈ 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 number of maxcliques in G0. Reference [4] contains several other characterizations, 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 ∈ 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 {x : 0 ≤ d(v, x) ≤ k} ⊆ 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. If every induced subgraph of G is disk-Helly, then G is called hereditary 18 TERRY A. MCKEE disk-Helly. References [3, 5] contain several characterizations of hereditary disk-Helly4 graphs, including the TERRY following A. MCKEE two: (i) being both chordal and clique-Helly, and (ii) being chordal with no induced Hajóssubgraph (where thedisk-HellyHajósgraph graphs,—sometimes including the called following the 3-sun two:—is (i) the being 3-ocular both chordal graph with and {uclique-Helly,1, u2, u3} independent, and (ii) being shown chordal as the with center no induced graph Hajóssubgraph in Figure 1). (where the Hajósgraph—sometimes called the 3-sun—is the 3-ocular graph with Corollaryu , u , u 5.independent,The following shown are asequivalent the center for graph every in chordal Fig. 1). graph G: { 1 2 3} (1) If Q is a maxclique of an induced subgraph G0 of G and |Q| ≥ 2, Corollary 5. The following are equivalent0 for every chordal0 graph G: then Q contains NG0 [v] ∩ NG0 [v ] for distinct v, v ∈ Q. (2)(5.1):EveryIf nontrivialQ is a maxclique maxclique of an of induced an induced subgraph subgraphG0 of ofGGandcontainsQ 2 a, then Q contains N [v] N [v ] for distinct v, v Q. | | ≥ simplicial edge. G0 ∩ G0 0 0 ∈ (3)(5.2):G is hereditaryEvery nontrivial disk-Helly. maxclique of an induced subgraph of G contains a simplicial edge. (4) G contains no induced Hajóssubgraph. (5.3): G is hereditary disk-Helly. Proof.(5.4):Since theG contains Hajósgraph no induced is the Hajóssubgraph.only chordal 3-ocular graph, the equiv- alencesProof. (1)Since⇔ (2) the andHajósgraph (1) ⇔ (4) is the follow only from chordal the 3-ocularp = 2 case graph, of Theorem the equiva- 2. Thelences equivalence (5.1) (5 (3).2) and⇔ (4) (5.1) is [5,(5 Thm..4) follow 1.2]. from the p = 2 case of Theorem 2. The equivalence⇔ (5.3) (5.4)⇔ is [5, Thm. 1.2]. 2 ⇔ 2. Maximal cographs and closed neighborhoods 2. Maximal cographs and closed neighborhoods In this section, cliques—which are simply the graphs that have no induced P3 subgraphs—willIn this section, cliques—which be generalized are to simply the complement-reducible the graphs that have no graphs induced(or P subgraphs—will be generalized to the complement-reducible graphs (or cographs3 )—which are the graphs that have no induced P4 subgraphs (or, equivalently,cographs)—which the graphs are the in graphs which that every have connected no induced inducedP4 subgraphs subgraph (or, has diameterequivalently, at most the graphstwo). See in which [1, 9] every for many connected additional induced characterizations. subgraph has Echoingdiameter the atCC most(G) notationtwo). See in [1,[1] for 9] for the many set of all additional inclusion-maximal characterizations. subsets Echoing the C(G) notation in [1] for the set of all inclusion-maximal subsets of V (G) thatC induce connected cographs, define a CC-subgraph of G to be a subgraphof V (G) ofthatG inducethat isc inducedonnected bycographs, a set in C defineC(G). a C-subgraph of G to be a subgraph of G that is induced by a set in C(G).C For each p ≥ 1, let Kp+4 − P4 denoteC the graph on the vertex set For each p 1, let Kp+4 P4 denote the graph on the vertex set w1,..., {w1, . . . , wp+2,≥ u1, u2} that− is complete except that edges w1u1, u{1u2 and uwwp+2do, u1 not, u2 occur.that is Equivalently, complete exceptK that− P edgesis thew1u chordal1, u1u2 (pand+ 2)-ocularu2w2 do not2 2 occur. Equivalently,} K P isp+4 the chordal4 (p + 2)-ocular graph with graph with vertices u , . . . , up+4 deleted.4 The p-clique {w , . . . , w } is the vertices u , . . . , u 3deleted.p+2− The p-clique w , . . . , w 3 is thep+2center of center of the3 K p+2− P graph. 3 p+2 the K P pgraph.+4 4 { } p+4 − 4 w w w 1 2 !! 3 aa BB ¢¢ @ !! @ aa u2 B ¢ u1 u2 w1 w2 !u1 b " PP !! b B ¢"" PP@ ! w3 w4

Figure 2. The graphs K5 P4 and K6 P4. Figure 2. The graphs K5 −−P4 and K6 − P4.

Lemma 6. For every graph G and p 1, a C-subgraph H contains ≥ C LemmaNG[v1] 6. ForNG every[vp] for graph distinctG andv1, . .p . ,≥ vp 1in,V a (CHC)-subgraphif and onlyH if containsH does not contain∩ · · · the ∩ center of a K P subgraph of G. NG[v1] ∩ · · · ∩ NG[vp] for distinctp+4 −v1, .4 . . , vp in V (H) if and only if H does not contain the center of a Kp+4 − P4 subgraphp of G. Proof. Some C-subgraph of G contains i=1NG[vi] for some set S = C p v1, . . . , vp of p vertices if and only if i=1NG[vi] contains no induced P4 path{ a, b, c,} d, which in turn is equivalent toT no set S a, b, c, d inducing T ∪ { } CHARACTERIZING GRAPH CLASSES 19

Tp Proof. Some CC-subgraph of G contains i=1 NG[vi] for some set S = Tp {v1, . . . , vp} of p vertices if and only if i=1 NG[vi] contains no induced P4 path a, b, c, d, which in turn is equivalent to no set S ∪ {a, b, c, d} inducing a Kp+4 − P4 subgraph of G with w1 = b, w2 = c, w3 = v1, ... , wp+2 = vp, u1 = d, and u2 = a (in the notation in the definition of Kp+4 − P4). Theorem 7. The following are equivalent for every graph G and p ≥ 1: (1) If H is a CC-subgraph of an induced subgraph G0 of G, then H contains NG0 [v1] ∩ · · · ∩ NG0 [vp] for distinct v1, . . . , vp ∈ V (H). (2) G contains no induced Kp+4 − P4 subgraph. Proof. To prove (1) ⇒ (2), suppose p ≥ 1 and condition (2) fails; specifically, 0 ∼ suppose G has an induced subgraph G = Kp+4 − P4 on the vertex set {w1, . . . , wp+2, u1, u2} as described in the definition of Kp+4 − P4. Take H 0 0 to be the CC-subgraph of G that is obtained by deleting w1 from G , and 0 note that H contains the center {w3, . . . , wp+2} of G . Lemma 6 then implies Tp that H does not contain i=1 NG0 [vi] for distinct vertices v1, . . . , vp, and so condition (1) fails. To prove (2) ⇒ (1), suppose p ≥ 1 and condition (1) fails; specifically, suppose H is a CC-subgraph of an induced subgraph G0 of G such that H Tp contains distinct vertices v1, . . . , vp without containing i=1 NG0 [vi]. Lemma 0 6 implies that H contains the center of a Kp+4 − P4 subgraph of G , and so condition (2) fails. A graph G is called ptolemaic if G is both chordal and contains no induced K5 − P4 subgraph (often called a gem); see [1] for additional characterizations. Corollary 8 corresponds to Theorem 4. Corollary 8. For every chordal graph G, every CC-subgraph H of an in- 0 duced subgraph G of G contains NG0 [v] for some v ∈ V (H) if and only if G is ptolemaic.

Proof. This follows from the p = 1 case of Theorem 7.

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Department of Mathematics and Statistics, Wright State University, 3640 Col. Glenn Highway, Dayton, Ohio 45435, USA E-mail address: [email protected]