On the well-coveredness of square graphs

Abstract

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 K1K1 or Kr,rKr,r for some r≥1r≥1. Moreover, we obtain a characterization of the graphs whose squares are well-covered in the case α(G)=α(G2)+kα(G)=α(G2)+kαG=αG2+kα(G)=α(G)2+k for $kin {0,1}$.