On k-independence in graphs with emphasis on trees

In a graphG= (V ,E) of order n and maximum degree , a subset S of vertices is a k-independent set if the subgraph induced by S has maximum degree less or equal to k − 1. The lower k-independence number ik (G) is the minimum cardinality of a maximal k-independent set in G and the k-independence number k(G) is the maximum cardinality of a k-independent set. We show that ik n − + k − 1 for any graph and any k , and i2 n − if G is connected, that k(T ) kn/(k + 1) for any tree, and that i2 (n + s)/2 2 for any connected bipartite graph with s support vertices. Moreover, we characterize the trees satisfying i2 = n− , k = kn/(k + 1), i2 = (n+ s)/2 or 2 = (n+ s)/2. © 2007 Elsevier B.V. All rights reserved.