On image-critical edges in König–Egerváry graphs Original Research Article

The stability number of a graph G, denoted by image, is the cardinality of a stable set of maximum size in G. If image, then e is an image-critical edge, and if image, then e is a image-critical edge, where image is the cardinality of a maximum matching in G. G is a König–Egerváry graph if its order equals image. Beineke, Harary and Plummer have shown that the set of image-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 image-critical edges are also image-critical, and that they coincide in bipartite graphs. For König–Egerváry graphs, we characterize image-critical edges that are also image-critical. Eventually, we deduce that image holds for any tree T, and describe the König–Egerváry graphs enjoying this property, where image is the number of image-critical vertices and image is the number of image-critical edges.