Combinatorial properties of the family of maximum stable sets of a graph

The stability number α(G) of a graph G is the size of a maximum stable set of G, core(G) = ∩{S : S is a maximum stable in G}, and ξ(G) = |core(G)|. In this paper we prove that for a graph G without isolated vertices, the following assertions are true: (i) if ξ(G) ≤ 1, then G is quasi-regularizable; (ii) if G is of order n and α(G) > (n + k − 1)/2, for some k ≥ 1, then ξ(G) ≥ k + 1, and ξ(G) ≥ k + 2, whenever n + k − 1 is even. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that α(G) > n/2 implies ξ(G) ≥ 1. In the case of König-Egerváry graphs, i.e., for graphs enjoying α(G) + μ(G) = n, where μ(G) is the maximum size of a matching of G, we prove that |core(G)| > |N(core(G))| is a necessary and sufficient condition for α(G) > n/2. Moreover, for bipartite graphs without isolated vertices, ξ(G) ≥ 2 is equivalent to α(G) > n/2. We also show that Hall’s marriage Theorem is valid for König-Egerváry graphs, and, it is sufficient to check Hall’s condition only for one specific stable set, namely, for core(G).