The algorithmic complexity of mixed domination in graphs

A three-valued function f defined on the vertices of a graph G = ( V, E), f : V 4 {-I. 0. I }, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every 1~ t V, ,f(N[o]) > 1, where N[c] consists of I: and every vertex adjacent to 1’. The weight of a minus dominating function is f(V) = c f(u), over all vertices L: t V. The minus domination number of a graph G, denoted y--(G). equals the minimum weight of a minus dominating function of G. The upper minus domination number of a graph G. denoted T-(G), equals the maximum weight of a minimal minus dominating function of G. In this paper we present a variety of algorithmic results. We show that the decision problem corresponding to the problem of computing y(respectively, r) is NP-complete even when restricted to bipartite graphs or chordal graphs. We also present a linear time algorithm for finding a minimum minus dominating function in an arbitrary tree.