ON THE WELL-COVEREDNESS OF SQUARE GRAPHS

The square of a graph G is obtained from G by putting an edge between two distinct vertices whenever their distance in G is 2. A graph is well-covered if every maximal independent set in the graph is of the same size. In this paper, we investigate the graphs whose squares are well-covered. We first provide a characterization of the trees whose squares are well-covered. Afterwards, we show that a bipartite graph G and its square are well-covered if and only if every component of G is K-1 or K-r,K-r for some r >= 1. Moreover, we obtain a characterization of the graphs whose squares are well-covered in the case alpha(G) = alpha(G(2)) + k for k is an element of {0, 1}.