When is G² a König-Egerváry Graph?

The independence number of a graph G, denoted by α(G), is the cardinality of a maximum independent set, and μ(G) is the size of a maximum matching in G. If α(G) + μ(G) equals its order, then G is a König-Egerváry 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 two. G is a square-stable graph if it enjoys the property α(G) = α(G ²). In this paper we show that G ² is a König-Egerváry graph if and only if G is a square-stable König-Egerváry graph.