Discrete Mathematics

Discrete Mathematics 310 (2010) 3398 303 Contents lists available at ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Maximal cliques in {P 2 P 3, C }-free graphs S.A. Choudum, T. Karthick Department of Mathematics, Indian Institute of Technology Madras, Chennai-600 036, India a r t i c l e i n f o a b s t r a c t Article history: Received 28 January 2010 Received in revised form 19 July 2010 Accepted 12 August 2010 Available online 9 September 2010 Keywords: Chordal graphs C -free graphs Maximal cliques Chromatic number We prove a decomposition theorem for the class G of {P 2 P 3, C }-free graphs. This theorem enables us to show that (i) every graph G in G has at most n + 5 maximal cliques where n is the number of vertices in G, and (ii) for every G in G, χ(g), where χ(g) (ω(g)) is the chromatic (clique) number of G. 2010 Elsevier B.V. All rights reserved. 1. Introduction All our graphs are simple, finite and undirected. For a graph G, let n and m, respectively denote the number of vertices and number of edges in G. A clique in a graph G is a set of pairwise adjacent vertices in G, and is maximal if it is not contained in a larger clique. We denote the number of maximal cliques in a graph G by µ(g). Moon and Moser [12] proved that µ(g) 3 n 3, for any graph G, and observed that this bound is attained by the graph ( t K 3 3) c, where t is a multiple of 3. A class of graphs G is said to have few cliques [1] if µ(g) p(n), for every G G, where p(n) denotes a polynomial in n. In 1965, Fulkerson and Gross [7] obtained an interesting characterization of interval graphs. During the course of their proof, they observed that any chordal graph has at most n maximal cliques. Since then several more classes of graphs with few cliques have been identified. These results are summarized in Table 1. It follows that the Maximum Clique Problem and the Maximum Weight Clique Problem for these classes of graphs are solvable in polynomial time, where as for a general class of graphs these problems are well known to be NP-complete. Various complexity issues on graphs with few cliques are investigated in [15]. For (P 2 P 3 )-free graphs, Maximum Independent Set Problem and Independent Dominating Set Problem are solvable in polynomial time [11] but Minimum Dominating Set Problem [1] and Maximum Clique Problem [13] are known to be NP-complete. In this paper, we are concerned with the class of {P 2 P 3, C }-free graphs, which includes interesting subclasses such as split graphs, pseudo-split graphs, {2K 2, C }-free graphs and {3K 1, C }-free graphs. See Fig. 1 for examples of graphs which are (i) {P 2 P 3, C }-free, (ii) (P 2 P 3 )-free but contains an induced C, and (iii) C -free but contains an induced (P 2 P 3 ). We prove a decomposition theorem for the class G of {P 2 P 3, C }-free graphs. This theorem enables us to show that (i) every graph G in G has at most n + 5 maximal cliques, where n is the number of vertices in G, and (ii) for every G in G, χ(g), where χ(g) (ω(g)) is the chromatic (clique) number of G. These extend the results proved in [2,3]. Throughout the paper, P t, C t, K t, respectively denote the induced path, induced cycle, complete graphs on t vertices. For notations and terminology not defined here, we follow West [17]. If H is an induced subgraph of G, we write H G. If H is Corresponding author. E-mail addresses: sac@iitm.ac.in (S.A. Choudum), karthick@smail.iitm.ac.in (T. Karthick). 0012-365X/$ see front matter 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2010.08.005

S.A. Choudum, T. Karthick / Discrete Mathematics 310 (2010) 3398 303 3399 Fig. 1. H 1 is a {P 2 P 3, C }-free graph, H 2 is a (P 2 P 3 )-free graph but contains an induced C, and H 3 is a C -free graph but contains an induced (P 2 P 3 ). Table 1 Classes of graphs with few cliques. G Upper bound for µ(g) Chordal n [7] C -free cn 2, c is a constant [6] {2K 2, C }-free n [2] {P 5, C }-free n [8] 7n Planar 3 [1] K t -free, t 2 max{n, n t 2 } [1] 2 t 2 {K 1,3, K 1 + C }-free 2m [9] Graph with boxicity k (2n) k [16] a subgraph of a graph G and if v V(G), then N H (v) = N G (v) V(H). If F is a family of graphs, G is said to be F -free if it contains no induced subgraph isomorphic to any graph in F. If S and T are two vertex disjoint subsets, then [S, T] denotes the set of edges with one end in S and the other in T. [S, T] is said to be complete if every vertex in S is adjacent with every vertex in T. [S] denotes the subgraph induced by S. The length of a shortest path between two vertices x, y is denoted by d(x, y), and d(x, S) := min{d(x, y) : y S}. For any integer i 0, N i (S) := {y V(G) : d(x, y) = i, for some x S}, and we simply denote it by N i. N 1 ({x}) is denoted by N(x), and N[x] = N(x) {x}. For a graph G, we denote a partition of V(G) into k ( 1) sets by (V 1, V 2,..., V k ). For a graph G, G c denotes the complement of G. If G 1 and G 2 are two vertex disjoint graphs, then their union G 1 G 2 is the graph with V(G 1 G 2 ) = V(G 1 ) V(G 2 ) and E(G 1 G 2 ) = E(G 1 ) E(G 2 ). Similarly, their join G 1 + G 2 is the graph with V(G 1 + G 2 ) = V(G 1 ) V(G 2 ) and E(G 1 + G 2 ) = E(G 1 ) E(G 2 ) {(x, y) : x V(G 1 ), y V(G 2 )}. For any positive integer k, kg denotes the union of k disjoint graphs each isomorphic with G. For a graph G, let χ(g) (ω(g)) denote its chromatic (clique) number. Next, we define an operation to combine various graphs. Let G be a graph on n vertices v 1, v 2,..., v n, and let H 1, H 2,..., H n be n vertex disjoint graphs. Then an expansion G(H 1, H 2,..., H n ) of G is the graph obtained from G by (i) replacing the vertex v i of G by H i, i = 1, 2,..., n, and (ii) joining the vertices x H i, y H j iff v i and v j are adjacent in G. An expansion is also called a composition; see [17]. If H i s are complete, it is called a complete expansion of G, and is denoted by K[G] or K[G](m 1, m 2,..., m n ) if H i = K mi. It is shown by Lovász [10] that if G, H 1, H 2,..., H n are perfect, then G(H 1, H 2,..., H n ) is perfect. So, if G is perfect, then K[G] is perfect. It is easily verified that if G is chordal, then K[G] is chordal, so in particular G + K t (t 0) is chordal. 2. The class of {P 2 P 3, C }-free graphs 2.1. Structure of {P 2 P 3, C }-free graphs In this section, we prove a decomposition theorem for a non-chordal {P 2 P 3, C }-free graph G which says that its vertex set V(G) can be partitioned into three subsets V 1, V 2 and V 3 such that [V 3 ] can be obtained as a complete expansion of one of the seventeen basic graphs shown in Fig. 2, and V 1 and V 2, respectively induce an edgeless graph and a complete graph. Theorem 1. A connected graph G that contains an induced C 6 is {P 2 P 3, C, C 5 }-free if and only if there exists a partition (V 1, V 2, V 3 ) of V(G) such that (1) [V 1 ] = K c m, for some m 0, (2) [V 2] = K t, for some t 0, (3) [V 3 ] = C 6, () [V 1, V 3 ] =, and (5) [V 2, V 3 ] is complete. Proof. Observe that if such a partition exists, then G is a {P 2 P 3, C, C 5 }-free graph that contains an induced C 6. To see the reverse implication, let us assume that [N 0 ] := [{v 1, v 2, v 3, v, v 5, v 6 }] = C 6 G. By assumptions on G, any x N 1 is adjacent to all the vertices of [N 0 ]. Then (i) [N 1 ] is complete (else, C G). (ii) N 2 is independent (else, P 2 P 3 G). (iii) N r =, if r 3 (else, P 2 P 3 G). It follows that (N 2, N 1, N 0 ) is a required partition of V(G).

300 S.A. Choudum, T. Karthick / Discrete Mathematics 310 (2010) 3398 303 Fig. 2. Basic graphs used in Theorem 2. Theorem 2. If G is a connected {P 2 P 3, C }-free graph, then G is either chordal or there exists a partition (V 1, V 2, V 3 ) of V(G) such that (1) [V 1 ] = K c m, for some m 0, (2) [V 2] = K t, for some t 0, (3) [V 3 ] is isomorphic to a graph obtained from one of the basic graphs G t (1 t 17) shown in Fig. 2 by expanding each vertex indicated in circle by a complete graph (of order 1), () [V 1, V 3 ] =, and (5) [V 2, V 3 \ S] is complete (see Fig. 2 for the set S). Proof. Let G be a connected {P 2 P 3, C }-free graph and for 1 t 17, let G t denote the class of graphs obtained from G t by the operations stated in the theorem. If G is {C 5, C 6 }-free, then G is chordal. If G is C 5 -free and contains an induced C 6, then a required partition is given by Theorem 1 where [V 3 ] = C 6 G 1 and S =. Hence, assume that G contains an induced C 5. Among all the induced 5-cycles in G, choose one, say C such that the number of vertices in V(G) \ V(C) that are adjacent to exactly two consecutive vertices of C is maximum. Let C = [N 0 ] = [{v 1, v 2, v 3, v, v 5 }] = C 5 G and let A = {x N 1 : [N(x) N 0 ] = K 1 }, B = {x N 1 : [N(x) N 0 ] = K 2 }, D = {x N 1 : [N(x) N 0 ] = P 3 }, F = {x N 1 : [N(x) N 0 ] = C 5 }. Then it is easily verified that N 1 = A B D F. For convenience, we further partition A, B and D as follows: A i = {x A : N(x) N 0 = {v i }}, B i = {x B : N(x) N 0 = {v i, v i+1 }}, D i = {x D : N(x) N 0 = {v i 1, v i, v i+1 }}, where 1 i 5, i mod 5. Then A = 5 i=1 A i, B = 5 i=1 B i, and D = 5 i=1 D i, and the following hold: (R1) N j =, for all j 3 (else, P 2 P 3 G); so V(G) = N 0 N 1 N 2. (R2) [A B D, N 2 ] = (else, P 2 P 3 G). For every i, 1 i 5, i mod 5, we have: (R3) (i) A i 1 (else, P 2 P 3 G). (ii) [A i, A i+1 ] = (else, C G).

S.A. Choudum, T. Karthick / Discrete Mathematics 310 (2010) 3398 303 301 (iii) [A i, A i+2 ] is complete (else, P 2 P 3 G). (iv) [A] is an induced subgraph of C 5 (a consequence of (i), (ii) and (iii)). (R) (i) B i 1 (else, P 2 P 3 G). (ii) [B i, B i+1 ] is complete (else, P 2 P 3 G). (iii) If B i, then B i+2 = = B i 2 (else, P 2 P 3 or C G). (iv) If B = 2, then B k, B k+1, for some k, 1 k 5, k mod 5 (a consequence of (iii)). (R5) (i) [D i ] is complete (else, C G). (ii) [D i, D i+2 ] = (else, C G). (iii) [D i, D i+1 ] is complete or there exist a unique d i D i and a unique d i+1 D i+1 such that d i d i+1 E(G) (else, P 2 P 3 or C G). In the latter case, D i+3 = (else, P 2 P 3 G). (iv) There exists a j, 1 j 5, j mod 5, such that [N 0 D] = K(C 5 ) Z, where Z {(d j, d j+1 ), (d, j+1 d j+2)} with d j D j, d j+1, d j+1 D j+1 (d j+1 d j+1 ) and d j+2 D j+2 ( a consequence of (i), (ii) and (iii)). (R6) (i) If A i, then B j =, for all j i + 2 (else, P 2 P 3 or C G). (ii) [A i, B i+2 ] is complete (else, P 2 P 3 G). (R7) (i) [A i, D i ] is complete (else, P 2 P 3 G). (ii) [A i, D i 1 ] = = [A i, D i+1 ] (else, P 2 P 3 G). (iii) If A i, then D i 2 = = D i+2 (else, P 2 P 3 or C G). (iv) If A i, then [D i, D i+1 ] and [D i, D i 1 ] are complete (else, P 2 P 3 G, by (i) and (ii)). (R8) (i) [B i, D j ] is complete, for all j i 2 (else, P 2 P 3 G). (ii) [B i, D i 2 ] = (else, C G). (iii) If B i, then either D i 1 = or D i+2 = (else, P 2 P 3 or C G). (iv) If B i, then [D i+1, D i+2 ] and [D i, D i 1 ] are complete (else, C G). (R9) [F] is complete (else, C G). (R10) [D i, F] is complete (else, C G). Hence, [N 0 D, F] is complete. (R11) N 2 is independent (else, P 2 P 3 G). (R12) (i) If x, y V(G) \ V(C) are such that xy E(G), and if there exists an index j (1 j 5, j mod 5) such that xv j, yv j 2 E(G) and xv j 2, yv j, xv j 1, yv j 1 E(G), then [{x, y}, F] is complete (else, P 2 P 3 or C G). (ii) If x, y A B and xy E(G), then N(x) F = N(y) F (else, C G). If we define V 1 = N 2, V 2 = F and V 3 = N 0 A B D, then the partition (V 1, V 2, V 3 ) of V(G) satisfies the following requirements of the theorem: (1) by (R11), (2) by (R9), () by (R2), and (5) by (R10) and (R12). So, the theorem is proved if we show that [V 3 ] satisfies (3). We will consider three cases depending on B. Note that B 2, by (R). Case 1: B = 0. If A =, we claim that the set Z defined in (R5(iv)) is empty. If not, (w. l. o. g.) assume that (d 1, d 2 ) Z. Then C := [{v 1, d 2, v 3, v, v 5 }] is an induced 5-cycle in G such that d 1 is adjacent to exactly two consecutive vertices v 1 and v 5 of C, a contradiction to the choice of C. So, by properties (R3), (R5) and (R7), we conclude that [V 3 ] = K(C 5 ) G 2, if A =, and [V 3 ] 9 G i=3 i, if A. Case 2: B = 1. Assume (w.l.o.g.) that B = {b 1 }. Then by (R6(i)), A i =, for all i {1, 2, 3, 5}. So, A = A 1. Also, by (R8(iii)), one of D 3 or D 5 is empty. Assume (w.l.o.g.) that D 5 =. Next, we claim that Z =. Assume to the contrary that Z. By (R8(iv)), Z = {(d 1, d 2 )} or Z = {(d 3, d )}. If Z = {(d 1, d 2 )}, then C := [{v 1, d 2, v 3, v, v 5 }] is an induced 5-cycle in G such that d 1 is adjacent to v 1 and v 5 of C, and b 1 is adjacent to v 1 and d 2 of C, a contradiction to the choice of C, and if Z = {(d 3, d )}, we get a similar contradiction. So, if A = 0, then by properties (R) and (R8), we see that [V 3 ] G 10 G 11. If A = 1, let A = A = {a }. By (R7(iii)), D 1 = = D 2. By (R6(ii)), [A, B 1 ] is complete. Then by using (R7(ii)) and (R8(i)), we see that D 3 = (else, C G). Since [A, D ] is complete (by R7(i)), these observations imply that [V 3 ] G 12. Case 3: B = 2. Assume (w.l.o.g.) that B = {b 1, b 2 }. Then by (R6(i)), A =. By (R8(iii)), we deduce that D 5 = or D 3 =, since B 1, and D 1 = or D =, since B 2. Hence one of the four sets D 1 D 5, D 5 D, D 3 D 1 or D D 3 is empty. So, if Z =, then since b 1 b 2 E(G) (by R(ii)), we see that one of D = or D 5 = (else, C G). Hence, by properties (R), (R5) and (R8), we conclude that [V 3 ] G 13 G 1 G 15, if D 1 D 5 = or D D 3 = or D 3 D 1 =, and [V 3 ] G 13 G 1 G 16, if D D 5 =. If Z, then by (R8(iv)), Z = {(d, d 5 )}. Now, we claim that D \ {d } = = D 5 \ {d 5 }. Assume to the contrary that d D \ {d }. Then b 1 b 2, d d 5, b 1 d 5, b 2 d E(G) (by R(ii), R5(iii), and R8(i)) and b 1d, b 2d 5 E(G) (by R8(ii)). But then [b 3, b, d, 1 d 2] = C G, a contradiction. So, D \ {d } =. Similarly, D 5 \ {d 5 } =. Thus, by properties (R), (R5) and (R8), and by the nature of Z, we conclude that [V 3 ] G 17. Corollary 1. If G is a connected {P 2 P 3, C }-free graph, then there exists a vertex v in G such that [N[v]] is chordal. Proof. If G is chordal, then [N[v]] is chordal, for all v V(G). If G is not chordal, then consider the partition (V 1, V 2, V 3 ) of V(G) described in Theorem 2. Let v V 3. It can be directly verified that [N Gj [v]] is chordal, for all j, 1 j 17. Moreover, [N G (v) V 2 ] is a complete graph. Thus, [N G (v)] is a complete expansion of [N Gj [v]] + K 1, where 1 j 17, and hence it is chordal. It follows that if G is a {P 2 P 3, C }-free graph (connected or disconnected), then there exists a vertex v such that [N[v]] is chordal.

302 S.A. Choudum, T. Karthick / Discrete Mathematics 310 (2010) 3398 303 2.2. Number of maximal cliques A universal vertex of a graph G is a vertex which is adjacent to all other vertices in G. For later use, we make the following simple observations and state a result due to Fulkerson and Gross [7]. (M1) µ(g 1 G 2 ) = µ(g 1 ) + µ(g 2 ). So, µ(g) = µ(g 1 ) + µ(g 2 ) + + µ(g k ), if G 1, G 2,..., G k are the components of G. (M2) µ(g 1 + G 2 ) = µ(g 1 ) µ(g 2 ). So, if T is a set of universal vertices in a graph G, then µ(g T) = µ(g). (M3) µ(k[g]) = µ(g). (M) If each vertex in G belongs to at most k maximal cliques, then µ(g) kn 2. Theorem A ([7]). If G is a chordal graph, then µ(g) n. Equality holds if and only if G has no edges. Theorem 3. If G is a {P 2 P 3, C }-free graph, then µ(g) n + 5. Proof. If G is chordal, then µ(g) n, by Theorem A. So, assume that G is not chordal. If G is connected, then V(G) admits a partition (V 1, V 2, V 3 ) as described in Theorem 2. Let T denote the set of all universal vertices in [V 2 V 3 ]. Note that T V 2. Then: µ(g) µ(g V 1 ) + V 1, since [V 1 ] is edgeless and every v V 1 is simplicial = µ(g V 1 T) + V 1, by (M2) = µ(g i ) + V 1, for some i, 1 i 17, by (M3) n + 5, since µ(g i ) V(G i ) + 5, for all i, 1 i 17. Next, assume that G is disconnected. Let F 1, F 2,..., F k denote the components of G. Since G is not chordal, at least one component, say F 1 admits a decomposition (V 1, V 2, V 3 ) of V(F 1 ) as given in Theorem 2 such that C 5 [V 3 ] or C 6 = [V3 ]. Since G is (P 2 P 3 )-free, for every i, 2 i k, F i is an edgeless graph and hence it has exactly one vertex. So, µ(g) = µ(f 1 ) + n V(F 1 ), by (M1) V(F 1 ) + 5 + n V(F 1 ), by the above argument = n + 5. The upper bound given in Theorem 3 is attained for the graphs P K c t, where t 0 and P is the Petersen graph. Corollary 2. If G is a {K 1 + (P 2 P 3 ), K 1 + C }-free graph, then µ(g) n2 +5n. 2 Proof. Note that [N(x)] is {P 2 P 3, C }-free, for any x V(G). So, by Theorem 3, any x V(G) belong to at most n + 5 maximal cliques. Hence the result follows by (M). If G is a {K 1 + (P 2 P 3 ), K 1 + C }-free graph, then the upper bound for µ(g) in Corollary 2 has to be quadratic in n. For example, consider the graph H = C 5 (K c, t K c, t K c, t K c, t K c t ), which is {K 1 + (P 2 P 3 ), K 1 + C }-free and has 5t vertices. Since maximal cliques in H are edges, we have µ(h) = 5t 2. An O(n 2 m)-time algorithm to generate all the maximal cliques in a graph which repeatedly contains a vertex whose neighborhood is chordal is described in Section 2 of [5]. Therefore, by Corollary 1, we have the following result. Theorem. All the maximal cliques in a {P 2 P 3, C }-free graph can be generated in O(n 2 m)-time. 2.3. Chromatic bound Blászik et al. [2] proved that for every {2K 2, C }-free graph G, χ(g) ω(g) + 1. It is shown in [3] that if G is a {3K 1, C }- free graph, then χ(g). We prove that this latter bound holds for the class of {P 2 P 3, C }-free graphs, which is a superclass of both {2K 2, C }-free graphs and {3K 1, C }-free graphs. This bound is optimal; see [3] for optimal graphs. A graph G is said to be perfect if χ(h) = ω(h), for every induced subgraph H of G. The strong perfect graph theorem (SPGT) [] states that G is perfect if and only if G is {C 2k+1, C c 2k+1 }-free, for every k 2. The following assertions are easy consequences of SPGT []. Lemma 1. If G is a {P 2 P 3, C, C 5 }-free graph, then G is perfect. Lemma 2. If G is a graph which admits a partition (V 1, V 2, V 3 ) of V(G) such that (1) [V 1 ] is an edgeless graph, (2) [V 2 ] is complete, (3) [V 3 ] is perfect, () [V 1, V 3 ] =, and (5) [V 2, V 3 ] is complete, then G is perfect. Theorem 5. If G(V, E) is a {P 2 P 3, C }-free graph, then χ(g). Proof. Assume that G is connected. We use Theorem 2 to deduce the upper bound. If G is chordal, then G is perfect and so χ(g) = ω(g). Else, there exists a partition (V 1, V 2, V 3 ) of V(G) as described in Theorem 2.

S.A. Choudum, T. Karthick / Discrete Mathematics 310 (2010) 3398 303 303 If [V 3 ] G j, where 1 j 17, j 2, let T j G j be the independent set defined in Fig. 2. It is easily verified that G j T j is a perfect graph. So, [V 3 \T j ] is a complete expansion of a perfect graph. Hence, by Lemmas 1 and 2, G T j is perfect. Therefore, χ(g) χ(g T j ) + χ([t j ]) ω(g) + 1. Next suppose that [V 3 ] G 2. Then, [V 3 ] = K[C 5 ](m 1, m 2, m 3, m, m 5 ), where m i 1, for all i, 1 i 5; let m 1 = min{m i : 1 i 5}. Let V = 3 V(K[C 5](m 1, m 1, m 1, m 1, m 1 )). Then [V 3 V ] 3 is perfect. So, (V 1, V 2, V 3 V ) 3 is a partition of V(G V ) 3 as described in Lemma 2. Hence, G V 3 is perfect. Since ω([v ]) = 3 2m 1 and χ([v 1 V ]) = 5m 1 3 2 (see [3]), we deduce that χ(g) χ([v (V 1 V )]) + 3 χ([v 1 V ]) 3 ω([v (V 1 V )]) + 5m1 3, since [V (V 1 V 3 2 )] G V, 3 and G V 3 is perfect 5m1 (ω(g) 2m 1 ) + 2, since 2m 1 ω(g). Acknowledgements The authors thank the referees for their valuable suggestions and comments. The second author acknowledges the CSIR, India for the financial support to carry out this research work. References [1] A.A. Bertossi, Dominating sets for split and bipartite graphs, Information Processing Letters 19 (198) 37 0. [2] Z. Blázsik, M. Hujter, A. Pluhár, Z. Tuza, Graphs with no induced C and 2K 2, Discrete Mathematics 115 (1993) 51 55. [3] S.A. Choudum, M.A. Shalu, The class of {3K 1, C }-free graphs, Australasian Journal of Combinatorics 32 (2005) 111 116. [] M. Chudnovsky, P. Seymour, N. Robertson, R. Thomas, The strong perfect graph theorem, Annals of Mathematics 16 (1) (2006) 51 229. [5] M.V.G. da Silva, K. Vušković, Triangulated neighborhoods in even-hole-free graphs, Discrete Mathematics 307 (2007) 1065 1073. [6] M. Farber, On diameters and radii of bridged graphs, Discrete Mathematics 73 (1989) 29 260. [7] D.R. Fulkerson, O. Gross, Incidence matrices and interval graphs, Pacific Journal of Mathematics 15 (1965) 835 855. [8] J.L. Fouquet, V. Giakoumakis, F. Maire, H. Thuillier, On graphs without P 5 and P 5, Discrete Mathematics 16 (1995) 33. [9] T. Kloks, K 1,3 -free and W -free graphs, Information Processing Letters 60 (1996) 221 223. [10] L. Lovász, A characterization of perfect graphs, Journal of Combinatorial Theory Series B 13 (1972) 95 98. [11] V.V. Lozin, R. Mosca, Independent sets in extensions of 2K 2 -free graphs, Discrete Applied Mathematics 16 (2005) 7 80. [12] J.W. Moon, L. Moser, On cliques in graphs, Israel Journal of Mathematics 3 (1965) 23 28. [13] S. Poljak, A note on stable sets and colorings of graphs, Commentationes Mathematicae Universitatis Carolinae 15 (197) 307 309. [1] E. Prisner, Graphs with few cliques, in: Proceedings of the 7th Quadrennial International Conference on the Theory and Applications of Graphs, 1992, in: Graph Theory, Combinatorics, and Applications, 1995, pp. 95 956. [15] B. Rosgen, L. Stewart, Complexity results on graphs with few cliques, Discrete Mathematics and Theoretical Computer Science 9 (2007) 127 136. [16] J.P. Spinrad, Efficient graph representations, in: Fields Institute Monographs, vol. 19, American Mathematical Society, 2003. [17] D.B. West, Introduction to Graph Theory, 2nd Edition, Prentice-Hall, Englewood Cliffs, New Jersey, 2000.