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 forms 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. For König-Egerváry graphs, we characterize μ-critical edges that are also α-critical. Eventually, we deduce that α (T) = ξ (T) + η (T) holds for any tree T, and describe the König-Egerváry graphs enjoying this property, where ξ (G) is the number of α-critical vertices and η (G) is the number of α-critical edges.