Some Classes of Strongly Perfect Graphs G

Discrete Mathematics 206 (1999) 197–203 www.elsevier.com/locate/disc Some classes of strongly perfect graphs G. Ravindra Regional Institute of Education Mysore, 570 006, India

Received 26 June 1996; accepted 9 September 1998

Abstract

A graph is perfect if for each of its induced subgraphs H, the chromatic number of H is equal to the maximum number of mutually adjacent vertices in it. A graph is ‘strongly perfect’ if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in it. Every strongly perfect graph is perfect but the converse is generally not valid. For example, the complement of an even cycle of length at least 6 is not strongly perfect though it is perfect. The strongly perfect graphs form an interesting class of perfect graphs, because the complement of a strongly perfect graph is not necessarily strongly perfect, unlike the case with the perfect graphs and their relevance to the famous Berge’s strong perfect graph conjecture which has been eluding a solution for more than three decades. Meyniel graphs, line graphs that are free from some graphs, comparability graphs, costrongly perfect graphs are some of the most important classes of strongly perfect graphs. Here, we summarize the results concerning strongly perfect graphs. c 1999 Published by Elsevier Science B.V. All rights reserved

We consider only simple graphs. A graph is perfect [2] if for each of its induced subgraphs H, the chromatic number of H is equal to the maximum number of mutually adjacent vertices in it. The perfect graphs were discovered by Berge during the early 1960. Berge [3] conjectured that a graph is perfect i it does not contain an odd cycle of length at least 5 or its complement as an induced subgraph. A graph is called Berge [6] if it does not contain an odd cycle of length at least 5 or its complement as an induced subgraph. Thus, the conjecture of Berge can be stated as follows. A graph G is perfect i G is Berge. The conjecture is still an unsettled problem though it is solved for many classes of graphs [7–9,11,13–15,28–30]. A graph is ‘strongly perfect’ if each of its induced subgraphs H contains an independent set which meets all the cliques (maximal complete subgraphs) in H. The strongly perfect graphs were also ÿrst introduced by Berge at a Monday Seminar (MSH) Paris, 1978. In one of the Monday Seminars Berge posed the following problem: Is every perfect graph strongly perfect?

(However, the converse of this can easily be veriÿed. See [4] or [28].) C6 (or C2n; n¿3) was the ÿrst counterexample given by Ravindra to the above problem. The strongly perfect graphs form an interesting class of perfect graphs because of the following. 1. If a graph is perfect, then its complement is perfect (LovÃasz perfect graph theorem [12]). But, the complement of a strongly perfect graph is not necessarily perfect. (For example, C2n;n¿3, is perfect but not strongly perfect.) 2. In view of Ravindra’s conjecture [20] that every p-critical graph is sp-critical, the strongly perfect graphs are closely related to the solution of the famous unsettled Berge’s Perfect Graph Conjecture. (A graph G is p-critical (sp-critical) if G is not perfect (strongly perfect) but every proper induced subgraph of G is perfect (strongly perfect)). 3. A strongly perfect graph serves as one of the best mathematical model for a real situation where one would like to choose an optimal set of leaders from a given set of people [22]. Berge and Duchet [4], Ravindra [18,19], ChvÃatal [5] and Hoang [10], Ravindra and Basavayya [1,27] have obtained several interesting results on strongly perfect graphs, and they are all listed below as facts.

Fact 1 (Berge and Duchet [4]; Ravindra [18]). Every strongly perfect graph is perfect.

Fact 2 (Berge and Duchet [4]; Ravindra [18]). Every P4-free graph is strongly perfect.

Fact 3 (Berge [4]). Every triangulated graph is strongly perfect. (A graph is triangulated if every cycle of length at least 4 has a chord.)

Fact 4 (Berge [4]). Every comparability graph is strongly perfect. (A graph is a comparability graph if its edges are transitively orientable.)

Fact 5 (Berge [4]). A perfect graph G =(V; E) is strongly perfect i for every in- 0 duced subgraph H of G; no two families C =(C1;C2;:::;Ck ) and D =(D1;D2;:::; Dk ) of maximal complete subgraphs of H (with possible repeated complete subgraphs) sat- isfy |C| = |D| and |C(v)|¿|D(v)| for all v ∈ V . (C(v) is the subfamily of the cliques of C which contain v. D(v) has the similar meaning.)

Fact 6 (Ravindra [19]). Every Meyniel graph is strongly perfect. (If every odd cycle of length at least ÿve in a graph G has atleast two chords; then G is called a Meyniel graph.)

Fact 7 (Ravindra [18]). The line graph L(G) of a graph G is strongly perfect if and only if each of the following properties is true.

(i) Every block of G is either bipartite or K4 − e or Kp (36p64). G. Ravindra / Discrete Mathematics 206 (1999) 197–203 199

Fig. 1.

(ii) If Cr and Cs are two even cycles such that V (Cr) ∩ V (Cs) 6= ∅; then |V (Cr) ∩ V (Cs)| is even. (iii) If Ci and Cj are two disjoint even cycles in G; then all the paths in G con- necting Ci and Cj are of odd length.

Fact 8 (Ravindra [18]). A line graph L(G) of G is strongly perfect i it does not contain C2n+1 (n¿2); or any one of the graphs in Fig. 1 as an induced subgraph.

Fact 9 (Rao and Ravindra [16]; Ravindra [18]). For a total graph T(G) the following properties are equivalent: (i) T(G) is strongly perfect. (ii) T(G) is perfect.

(iii) Every block of G is either K2 or K3.

Fact 10 (Ravindra [21]). Every strongly perfect B-graph G contains a maximum independent set which meets all the maximal complete subgraphs of G. (A graph G is a B-graph if every vertex of G is in a maximum independent set of G.) 200 G. Ravindra / Discrete Mathematics 206 (1999) 197–203

Fact 11 (ChvÃatal [5]). Every perfectly orderable graph is strongly perfect. (A graph is perfectly orderable if its vertex set admits a linear order ¡ such that no induced

P4: abcd has a¡b; d¡c:)

Fact 12 (Basavayya and Ravindra [1]). A line graph L(G) of a (0; 1)-graph G is strongly perfect i it has no C2n+1;n¿2; as an induced subgraph. (The (0; 1)- graph of a (0; 1)-matrix A is the graph G(A) whose vertices are in 1–1 corre- spondence with the set of 1’s in A and two vertices are adjacent in G(A) i the corresponding 1’s lie in the same row or same column. G(A) is the (0; 1)-graph of A.)

Fact 13 (Ravindra [27]). For a Cartesian product G of non-trivial graphs; the following properties are equivalent. (i) G is strongly perfect. (ii) G is very strongly perfect. (iii) G is bipartite. (iv) G is perfectly orderable.

(A graph G is very strongly perfect if for each induced subgraph H of G; every vertex of H is in an independent set of H which meets all the maximal complete subgraphs of H.)

Fact 14 (Ravindra [27]). For a tensor product graph G; the following are equivalent. (i) G is very strongly perfect. (ii) G is strongly perfect. (iii) G is a bipartite graph. (iv) G is perfectly orderable.

Fact 15 (Hoang [10]). G is very strongly perfect i G is Meyniel.

Although Fact 5 gives a necessary and sucient condition for a graph to be strongly perfect, there is no characterization of strongly perfect graphs in terms of forbidden subgraphs (that is a complete set of sp-critical graphs is not known). However, some of the forbidden subgraphs are identiÿed by Ravindra (C2n, graphs of the Fig. 1), Berge (Fig. 2), ChvÃatal ((Fig. 3), personal communication to Ravindra), Ma ray (Fig. 4, personal communication to Ravindra).

Further, with respect to K1; 3-free graphs, Ravindra [19], has conjectured that C2n+1 (n¿2); Cn (n¿5) and the graphs of the Fig. 1 are the only sp-critical graphs. Lovasz’s perfect graph theorem asserts that the complement of a perfect graph is perfect. But the complement of a strongly perfect graph is not necessarily strongly perfect. An even cycle of length at least 6 is strongly perfect but its complement is not strongly perfect. So one would naturally be motivated to study those graphs G such that G and G are strongly perfect. A graph G is called costrongly per- G. Ravindra / Discrete Mathematics 206 (1999) 197–203 201

Fig. 2.

Fig. 3.

Fig. 4. fect if G and G are strongly perfect. Costrongly perfect graphs were studied by Ravindra and Basavayya [23–27] and the following are the results obtained by them.

Fact 16 (Ravindra [23]). Every triangulated graph is costrongly perfect. 202 G. Ravindra / Discrete Mathematics 206 (1999) 197–203

Fig. 5.

Fact 17 (Ravindra and Basavayya [25,26]). A nearly bipartite graph is costrongly perfect i it has no C2n;n¿3 as an induced subgraph.

Fact 18 (Ravindra [24]). A line graph G is costrongly perfect i it does not contain a cycle of length at least 5 or any one of the graphs of the Fig. 5 as an induced subgraph.

Fact 19 (Ravindra and Basavayya [27]). Ln(G) is costrongly perfect for every n¿1 if G = K1; 3;K3;C4 or Pk .

Fact 20 (Ravindra and Basavayya [23]). The following properties are equivalent for a total group T(G). (i) T(G) is costrongly perfect. (ii) T(G) is strongly perfect. (iii) T(G) is perfect.

(iv) Every block of G is either K2 or K3.

Fact 21 (Ravindra and Basavayya [27]). For a Cartesian product G of nontrivial graphs G1 and G2; the following properties are equivalent. (i) G is costrongly perfect.

(ii) GisC2n (n¿3)-free bipartite graph. (iii) G1 or G2 is K2 and the other is a tree.

Fact 22 (Ravindra and Basavayya [27]). A tensor product graph G is costrongly perfect i G is C2n (n¿3)-free bipartite graph.

References

[1] D. Basavayya, G. Ravindra, Strongly perfect line graphs of (0,1)-graphs, J. Math. Phys. Sci. 21 (4) (1987). [2] C. Berge, Les problÃemes de colorations en theeorie les graphes, Publ. Inst. Statist. Univ. Paris 9 (1960) 123–160. G. Ravindra / Discrete Mathematics 206 (1999) 197–203 203

[3] C. Berge, Sur une conjecture relative a.u. problÃeme des codes optimaux, Comom, 13ieme, Assemblee Generale de l’ URSI, Tokyo, 1962. [4] C. Berge, P. Duchet, in: C. Berge, V. ChvÃatal (Eds.), Strongly perfect graphs in topics on perfect graphs, Ann. Discrete Math., vol. 21, 1984. [5] V. ChvÃatal, Perfectly orderable graphs, in: C. Berge, V. ChvÃatal (Eds.), Topics on perfect graphs, Ann. Disc. Math., vol. 21, 1984, pp. 63–65. [6] V. ChvÃatal, Recent results on perfect graphs, Technical Report, Dept. of Computer Science, Rutgers University, 1987. [7] V. ChvÃatal, Sbihi, Bull-free graphs are perfect, Graphs Combin. 3 (1987) 127–139. [8] R.B. Hayward, Weakly triangulated graphs, J. Combin. Theory Ser. B 39 (1985) 200–208. [9] R.B. Hayward, Murky graphs are perfect, J. Combin. Theory Ser. B 49 (1990) 220–235; MR 91F: 05053. [10] C.T. Hoang, On a conjecture of meyniel, J. Combin. Theory Ser. B 42 (1987) 302–312. [11] S. Liping, Two classes of perfect graphs, Technical Report, Dept. of Computer Sci., Rutgers University, 1988; J. Combin. Theory Ser. B 53 (1991) 273–291. [12] L. LovÃasz, A characterization of perfect graphs, J. Combin. Theory Ser. B 13 (1972) 95–98. [13] S. Olariu, Strong perfect graph conjecture for pan-free graphs, J. Combin. Theory Ser. B 47 (1989) 187–191. [14] K.R. Parthasarathy, G. Ravindra, The strong perfect graph conjecture is true for K1; 3-free graphs, J. Combin. Theory Ser. B 21 (1976) 212–223. [15] K.R. Parthasarathy, G. Ravindra, The validity of strong perfect graph conjecture for (K4-e)-free graphs, J. Combin. Theory 13 (26) (1979) 98–100. [16] S.B. Rao, G. Ravindra, A characterization of perfect total graphs, J. Math. Phys. Sci. 11 (1) (1977) 25–26. [17] G. Ravindra, Inter relations among graph invariants, Ph.D. Thesis, IIT, Madras, 1976. [18] G. Ravindra, Strongly perfect line graphs and total graphs, Colloquia Mathematica Societatis, JaÃnos Bolyai, 37, Finite and Inÿnite Sets, Eger, Hungary, 1981, pp. 621–633. [19] G. Ravindra, Meyniel graphs are strongly perfect, J. Combin. Theory Ser. B 35 (2) (1982) 187–190. (Also in: C. Berge, V. ChvÃatal, Topics on Perfect Graphs, vol. 21, 1984, pp. 145–148.) [20] G. Ravindra, Research problems, Discrete Math. 80 (1990) 105–107. [21] G. Ravindra, Strongly perfect graphs, Invited Talk, Proc. 4th Internat. Conf. on Graph Theory and Combinatorics, Marseille, France, 9–13 July 1990. [22] G. Ravindra, Strongly perfect graphs, Invited Talk, 6th Annual Conf. of RMS, Rajkot, India, 1–3 June 1991. [23] G. Ravindra, D. Basavayya, Costrongly perfect graphs, J. Math. Phys. Sci. 22 (3) (1988) 439–444. [24] G. Ravindra, D. Basavayya, Costrongly perfect line graphs, Sankhya: The Indian J. Statist. Ser. A 54 (1991). [25] G. Ravindra, D. Basavayya, Costrongly perfect bipartite graphs, J. Math. Phys. Sci. 22 (3) (1992). [26] G. Ravindra, D. Basavayya, A characterization of nearly bipartite graphs with strongly perfect complements, J. Ramanujan Math. Soc. 9 (1) (1994) 79–87. [27] G. Ravindra, D. Basavayya, Strongly and costrongly perfect product graphs, J. Math. Phys. Sci. 29 (2) (1995) 79–80. [28] G. Ravindra, K.R. Parthasarathy, Perfect product graphs, Discrete Math. 20 (1977) 177–186. [29] A.C. Tucker, The validity of perfect graph conjecture for K4-free graphs, in: C. Berge, ChvÃatal (Eds.), Topics on Perfect Graphs, vol. 21, 1984, pp. 149–157. [30] A.C. Tucker, Colouring of (K4 − e)-free graphs, J. Combin. Theory Ser. B 42 (1987) 313–318.