Distance independent domination in graphs

Let n greater than or equal to 1 be an integer and let G be a graph of order p. A set I-n of vertices of G is n-independent if the distance between every two vertices of I-n is at least n + 1. Furthermore, I-n is defined to be an n-independent dominating set of G if I-n is an n-independent set in G and every vertex in V(G) - I-n is at distance at most n from some vertex in I-n. The n-independent domination number, i(n)(G). is the minimum cardinality among all n-independent dominating sets of G. Hence i(1)(G) = i(G) where i(G) is the independent domination number of G. We establish the existence of a connected graph G every spanning tree T of which is such that i(n)(T) < i(n)(G). For n is an element of {1,2} we show that, for any tree T and any tree T' obtained from T by joining a new vertex to some vertex of T, we have i(n)(T) 2 i(n)(T'). However we show that this is not true for n greater than or equal to 3. We show that the decision problem corresponding to the problem of computing i(n)(G) is NP-complete, even when restricted to bipartite graphs. Finally, we obtain a sharp lower bound on i(n)(G) for a graph G.