When G² is a Konig-Egervary graph?

The square of a graph G is the graph G^2 with the same vertex set as in G, and an edge of G^2 is joining two distinct vertices, whenever the distance between them in G is at most 2. G is a square-stable graph if it enjoys the property alpha(G)=alpha(G^2), where alpha(G) is the size of a maximum stable set in G. In this paper we show that G^2 is a Konig-Egervary graph if and only if G is a square-stable Konig-Egervary graph.