On α-Critical Edges in König-Egerváry 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 α(G − e) > α(G), then e is an α-critical edge, and if μ(G − e) < μ(G), then e is a μ-critical edge, where μ(G) is the cardinality of a maximum matching in G. G is a König-Egerváry graph if its order equals α(G) + μ(G). Beineke, Harary and Plummer have shown that the set of α-critical edges of a bipartite graph is a matching. In this paper we generalize this statement to König-Egerváry graphs. We also prove that in a König-Egerváry graph α-critical edges are also μ-critical, and that they coincide in bipartite graphs. Eventually, we deduce that α(T ) = ξ(T ) + η(T ) holds for any tree T , and characterize the König-Egerváry graphs enjoying this property, where ξ(G) is the number of α-critical vertices of G, and η(G) is the number of α-critical edges of G.