On [formula omitted]-critical edges in König–Egerváry graphs

Abstract

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.