On Konig-Egervary Square-Stable Graphs

The stability number of a graph G, denoted by α(G), is the cardinality of a maximum stable set, and µ(G) is the cardinality of a maximum matching in G. If α(G) + µ(G) equals its order, then G is a König-Egerváry graph. In this paper we deal with square-stable graphs, i.e., the graphs G enjoying the equality α(G) = α(G 2), where G 2 denotes the second power of G. In particular, we show that a König-Egerváry graph is square-stable if and only if it has a perfect matching consisting of pendant edges, and in consequence, we deduce that well-covered trees are exactly the square-stable trees.