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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 intersects.
By definition, G is a CIS-graph if and only if the complementary graph \(\overline {G}\) is a CIS-graph. 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 simple 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, \(v^{\prime }_1, \ldots , v^{\prime }_k,\) form a stable set S, and \((v_i,v^{\prime }_i)\) is an edge for all i = 1, …, k, and there are no other edges. The complementary graph \(\overline {S_k}\) 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. For k = 2 this observation was made by Claude Berge in 1985. However, these conditions are only necessary.
The following sufficient conditions are more difficult to prove: G is a CIS-graph whenever Gcontains no induced 3-combs and 3-anti-combs, and every induced 2-comb is settled in G, as it was conjectured by Vasek Chvatal in early 90s. First partial results were published by his student Wenan Zang from Rutgers Center for Operations Research. Then, the statement was proven by Deng, Li, and Zang. Here we give an alternative proof, which is of independent interest; it is based on some properties of the product of two Petersen graphs.
It is an open question whether G is a CIS-graph if it 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 \(\mathcal {G} = (V; E_1, \ldots , E_d)\) as a complete graph whose edges are colored by dcolors (that is, partitioned into d sets). We say that \(\mathcal {G}\) is a CIS-d-graph (has the CIS-d-property) if \(\bigcap _{i=1}^d C_i \neq \emptyset \) whenever for each i = 1, …, d the set Ci is a maximal color i-free subset of V , that is, (v, v′)∉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 \(\mathcal {G} = (V; E_1, \ldots , E_d)\) be a Gallai d-graph such that at least d − 1 of its d chromatic components are CIS-graphs, then \(\mathcal {G}\) has the CIS-d-property. In particular, the remaining chromatic component of \(\mathcal {G}\) is a CIS-graph too. Moreover, all 2d unions of d chromatic components of \(\mathcal {G}\) are CIS-graphs.