A Characterization of Almost CIS Graphs

A graph G is called CIS if each maximal clique intersects each maximal stable set in G, and is called almost CIS if it has a unique disjoint pair (C, S) consisting of a maximal clique C and a maximal stable set S. While it is still unknown if there exists a good structural characterization of all CIS graphs, in this note we prove the following Andrade-BorosGurvich conjecture: A graph is almost CIS if and only if it is a split graph with a unique split partition. Let G = (V, E) be a graph. A clique of G is a set of pairwise adjacent vertices, and a stable set of G is a set of pairwise nonadjacent vertices. We call G a CIS graph if each maximal clique intersects each maximal stable set in G, where the adjective maximal is meant with respect to set-inclusion rather than size. The study of CIS graphs dates back to the 1960s when Grillet [8] proved that in every partially ordered set containing no quadruple (a, b, c, d) such that a < b, c < d, b covers c, and the remaining three pairs of elements are incomparable, each maximal chain meets each maximal antichain. With an attempt to generalize this theorem, Berge [2] made a conjecture and posed a research problem in terms of CIS graphs; see [9] for their solutions. Later, Chvátal [4, 9] proposed another conjecture concerning CIS graphs as a variation on Berge’s problem, which was established independently by Andrade, Boros, and Gurvich [1] and Deng, Li, and Zang [5, 6]. We refer to [1] for an in-depth account of CIS graphs. Despite considerable research effort, it is still unknown if there exists a good structural characterization of all CIS graphs. In this regard, Chvátal [4, 9] suggested the following problem. Problem. How difficult is it to recognize CIS graphs? As pointed out by Andrade, Boros, and Gurvich [1], CIS graphs somehow resemble perfect graphs in several ways; they also conjectured that this recognition problem, though very difficult, is polynomial-time solvable. On the other hand, given a graph G together with a specified maximal stable set S, it is co-NP -complete [9] to decide if S intersects every maximal clique of G. By definition, if a graph G is not CIS, then it contains at least one disjoint pair (C, S) consisting of a maximal clique C and a maximal stable set S; such a (C, S) is called a non-CIS pair of G. Andrade, Boros, and Gurvich [1] proposed to call a graph almost CIS if it has a unique non-CIS pair, and discovered that almost CIS graphs are closely related to some well-known class of graphs. A graph is called split if its vertex set admits a partition (C, S), called a split partition, such that C is a clique and S is a stable set. As characterized by Foldes and Hammer [7], a graph is split if and only if it contains none of 2K2, C4, and C5 as an induced subgraph. Moreover, a split graph may have several split partitions; see, for instance, the graph obtained from a path abcd by adding a fifth vertex e and making it adjacent to both b and c. It was shown in [3] that Proposition 1. A split graph has more than one split partition if and only if it is CIS.