Three-valued k-neighborhood domination in graphs

Let k~l be an integer, and let G = (V, E) be a graph. The closed kneighborhood N k[V] of a vertex v E V is the set of vertices within distance k from v. A 3-valued function f defined on V of the form f : V --+ { -1,0, I} is a three-valued k-neighborhood dominating function if the sum of its function values over any closed k-neighborhood is at least 1. The weight of a threevalued k-neighborhood dominating function is f(V) L f( v), over all vertices v E V. The three-valued k-neighborhood domination number of a graph G, denoted Ik(G), equals the minimum weight of a three-valued k-neighborhood dominating function of G. For k ~ 2, we establish the existence of trees with three-valued k-neighborhood domination number less than any given negative number. We show that the decision problem corresponding to the problem of computing Ik is NP-complete even when restricted to bipartite graphs.