On unique independent sets in graphs

Abstract

For a nonnegative integer k, a subset I of the vertex set V( G) of a simple graph G is said to be k-independent if I is independent and every independent subset I′ of G with | I′|⩾| I|−( k−1) is a subset of I. A set I of vertices is called a strong k-independent set of G if I is k-independent and the set V( G)− I is independent in G. First we give several characterizations of k-independent sets for some classes of graphs. Then we characterize trees which have (strong) k-independent sets. Finally, we obtain lower bounds on the number of edges in graphs which have k-independent sets.