On [formula omitted]-independence in graphs with emphasis on trees

Abstract

In a graph G = ( V , E ) of order n and maximum degree Δ , 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 i k ( G ) is the minimum cardinality of a maximal k -independent set in G and the k -independence number β k ( G ) is the maximum cardinality of a k -independent set. We show that i k ⩽ n - Δ + k - 1 for any graph and any k ⩽ Δ , and i 2 ⩽ n - Δ if G is connected, that β k ( T ) ⩾ kn / ( k + 1 ) for any tree, and that i 2 ⩽ ( n + s ) / 2 ⩽ β 2 for any connected bipartite graph with s support vertices. Moreover, we characterize the trees satisfying i 2 = n - Δ , β k = kn / ( k + 1 ) , i 2 = ( n + s ) / 2 or β 2 = ( n + s ) / 2 .