On graphs whose maximal cliques and stable sets intersect

We say that a graph G has the CIS-property and call it a CIS-graph if every maximal clique and every maximal stable set of G intersect. By definition, G is a CIS-graph if and only if the complementary graph G is a CISgraph. Let us substitute a vertex v of a graph G′ by a graph G′′ and denote the obtained graph by G. It is also easy to see that G is a CIS-graph if and only if both G′ and G′′ are CIS-graphs. In other words, CIS-graphs respect complementation and substitution. Yet, this class is not hereditary, that is, an induced subgraph of a CIS-graph may have no CIS-property. Perhaps, for this reason, the problems of efficient characterization and recognition of CIS-graphs are difficult and remain open. In this paper we only give some necessary and some sufficient conditions for the CIS-property to hold. There are obvious sufficient conditions. It is known that P4-free graphs have the CIS-property and it is easy to see that G is a CIS-graph whenever each maximal clique of G has a simplicial vertex. However, these conditions are not necessary. There are also obvious necessary conditions. Given an integer k ≥ 2, a comb (or k-comb) Sk is a graph with 2k vertices k of which, v1, . . . , vk, form a clique C, while others, v′ 1, . . . , v ′ k, form a stable set S, and (vi, v ′ i) is an edge for all i = 1, . . . , k, and there are no other edges. The complementary graph Sk is called an anti-comb (or k-anti-comb). Clearly, S and C switch in the complementary graphs. Obviously, the combs and anti-combs are not CIS-graphs, since C ∩ S = ∅. Hence, if a CIS-graph G contains an induced comb or anti-comb then it must be settled, that is, G must contain a vertex v connected to all vertices of C and to no vertex of S. However, these conditions are only necessary. The following sufficient conditions are more difficult to prove: G is a CIS-graph whenever G contains no induced 3-combs and 3-anti-combs, and every induced 2comb is settled in G. It is an open question whether G is a CIS-graph if G contains no induced 4-combs and 4-anti-combs, and all induced 3-combs, 3-anti-combs, and 2-combs are settled in G. We generalize the concept of CIS-graphs as follows. For an integer d ≥ 2 we define a d-graph G = (V ;E1, . . . , Ed) as a complete graph whose edges are colored by d colors (that is, partitioned into d sets). We say that G is a CIS-d-graph (has the CIS-d-property) if ⋂d i=1 Ci 6= ∅ whenever for each i = 1, . . . , d the set Ci is a maximal color i-free subset of V , that is, (v, v′) 6∈ Ei for any v, v ′ ∈ Ci. Clearly, in case d = 2 we return to the concept of CIS-graphs. (More accurately, CIS-2-graph is a pair of two complementary CIS-graphs.) We conjecture that each CIS-d-graph is a Gallai graph, that is, it contains no triangle colored by 3 distinct colors. We obtain results supporting this conjecture and also show that if it holds then characterization and recognition of CIS-d-graphs are easily reduced to characterization and recognition of CIS-graphs. We also prove the following statement. Let G = (V ;E1, . . . , Ed) be a Gallai d-graph such that at least d − 1 of its d chromatic components are CIS-graphs, then G has the CIS-d-property. In particular, the remaining chromatic component of G is a CIS-graph too. Moreover, all 2 unions of d chromatic components of G are CISgraphs.