On ${\alpha }^{++}$-Stable Graphs

Abstract

The stability number of a graph G, denoted by alpha(G), is the cardinality of a stable set of maximum size in G. A graph is well-covered if every maximal stable set has the same size. G is a Koenig-Egervary graph if its order equals alpha(G) + mu(G), where mu(G) is the cardinality of a maximum matching in G. In this paper we characterize ${\alpha }^{++}$-stable graphs, namely, the graphs whose stability numbers are invariant to adding any two edges from their complements. We show that a König-Egerváry graph is ${\alpha }^{++}$-stable if and only if it has a perfect matching consisting of pendant edges and no four vertices of the graph span a cycle. As a corollary it gives necessary and sufficient conditions for ${\alpha }^{++}$-stability of bipartite graphs and trees. For instance, we prove that a bipartite graph is ${\alpha }^{++}$-stable if and only if it is well-covered and C4-free.

11 pages, 3 figures