On α-Stable König-Egervary Graphs

The stability number of a graph G, denoted by α(G), is the cardinality of a stable set of maximum size in G. If its stability number remains the same upon the addition of any edge, then G is called α-stable. G is a König-Egervary graph if its order equals α(G) + μ(G), where μ(G) is the cardinality of a maximum matching in G. In this paper we characterize α-stable König-Egervary graphs, generalizing some previously known results on bipartite graphs and trees. Namely, we prove that a König-Egervary graph G = (V,E) is α-stable if and only if either |∩{V − S : S ∈ Ω(G)}| = 0, or |∩{V − S : S ∈ Ω(G)}| = 1, and G has a perfect matching (where Ω(G) denotes the family of all maximum stable sets of G). Using this characterization we obtain several new findings on general König-Egervary graphs, for example, the equality |∩{S : S ∈ Ω(G)}| = |∩{V − S : S ∈ Ω(G)}| is a necessary and sufficient condition for a König-Egervary graph G to have a perfect matching.