On the integrity of distance domination in graphs

Let nand k be positive integers and let G be a graph. A set D of vertices of G is defined to be an (n, k )-dominating set of G if every vertex of V( G) D is within distance n from at least k vertices of D. The minimum cardinality among all (n, k )-dominating sets of G is called the (n, k )-domination number of G and is denoted by 'Yn,k(G). A set I of vertices of G is defined to be an (n, k)independent set in G if every vertex of I is within distance n from at most k-1 other vertices of I in G. We denote by f3n,k( G) the maximum cardinality of an (n, k)-independent set of G. We show that the problem of computing 'Yn,k is in the NP-complete class, even when restricted to bipartite graphs and chordal graphs. We prove that in every graph there exist some subsets of vertices that are both (n, k )-independent and (n, k )-dominating, so In,k :s; f3n,k' We also investigate lower and upper bounds on 'Yn,k' Australasian Journal of Combinatorics .!Q.( 1994) I pp. 29-43