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 set in G}, and ξ(G)=|core(G)|. In this paper we prove that for a graph G the following assertions are true: (i) if G has no isolated vertices, and ξ(G)1, then G is quasi-regularizable; (ii) if the order of G is n, and α(G)(n+k-min{1, |N(core(G))|})/2, for some k1, then ξ(G)k+1; moreover, if n+k-min{1,|N(core(G))|} is even, then ξ(G)k+2. The last finding is a strengthening of a result of Hammer, Hansen, and Simeone, which states that ξ(G)1 is true whenever α(G)n/2. In the case of König- Egerváry graphs, i.e., for graphs enjoying the equality 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. Furthermore, 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 true for König-Egerváry graphs, and, it is sufficient to check Hall's condition only for one specific stable set, namely for core(G).