Generalizations of Grillet's theorem on maximal stable sets and maximal cliques in graphs

A Characterization of Almost CIS Graphs

“…Since ac ∈ E, from (3) we see that a has at least one neighbor in N S (c). Note that D − {a} is contained in K − {d}, so all vertices in D − {a} are adjacent to d. Thus (9) holds.…”

“…On the other hand, given a graph G together with a specified maximal stable set S, it is co-N P -complete [9] to decide if S intersects every maximal clique of G.…”

“…Since (C, S) is the unique non-CIS pair of G, by (9) there exists a vertex e / ∈ D such that D ∪ {e} is a clique and T ∪ {e} is a stable set in G. In view of the edge eb, we have e / ∈ S. Since ea ∈ E and da / ∈ E, we get e = d. So ed / ∈ E as d ∈ T , which implies e / ∈ C. Therefore e ∈ A − (B ∪ {a}). Since e is adjacent to no vertex in N S (c), by (3) we obtain ec / ∈ E. Finally, observe that each vertex in K is adjacent to either e or c, so by Proposition 2 the maximal clique K is disjoint from any maximal stable set containing {e, c}, contradicting the hypothesis that (C, S) is the unique non-CIS pair of G.…”

“…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].…”

“…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.…”