On $\alpha $-Square-Stable Graphs

The stability number of a graph G, denoted by alpha(G), is the cardinality of a maximum stable set, and mu(G) is the cardinality of a maximum matching in G. If alpha(G) + mu(G) equals its order, then G is a Koenig-Egervary graph. We call G an $\alpha$-square-stable graph, shortly square-stable, if alpha(G) = alpha(G*G), where G*G denotes the second power of G. These graphs were first investigated by Randerath and Wolkmann. In this paper we obtain several new characterizations of square-stable graphs. We also show that G is an square-stable Koenig-Egervary graph if and only if it has a perfect matching consisting of pendant edges. Moreover, we find that well-covered trees are exactly square-stable trees. To verify this result we give a new proof of one Ravindra's theorem describing well-covered trees.