On k-independence in graphs with emphasis on trees

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