On the structure of α-stable graphs

The stability number α(G) of a graph G is the cardinality of a stability system of G (that is of a stable set of maximum size). A graph is α;-stable if its stability number remains the same upon both the deletion and the addition of any edge. Trying to generalize some stable trees properties, we show that there does not exist any α-stable chordal graph, and we prove that: if G is a connected bipartite graph, then the following assertions are equivalent: (i) G is α-stable; (ii) G can be written as a vertex disjoint union of connected bipartite graphs, each of them having exactly two stability systems covering its vertex set; (iii) G has perfect matchings and ∩{M: M is a perfect matching of G} = 0; (iv) for any vertex of G there are at least two edges incident to this vertex and contained in some perfect matchings; (v) any vertex of G belongs to a cycle, whose edges are alternately in and not in a perfect matching of G; and (vi) ∩{S: S is a stability system of G} = 0 ∩{M: M is a maximum matching of G}.