Minus domination in graphs

We introduce one of many classes of problems which can be defined in terms of 3-valued functions on the vertices of a graph G = (V, E) of the form f : V + { 1, 0, l}. Such a fknction is said to be a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every t’ E V, ,f(N[r~])> 1, where N[a] consists of 1: and every vertex adjacent to u. The weight of a minus dominating function is f(V) = c ,f(tl), over all vertices u E V. The minus domination number of a graph G, denoted v-(G), equals the minimum weight of a minus dominating function of G. For every graph G, y(G) 4, then y(T) y-(T) <(n 4)/S and this bound is sharp. We attempt to classify graphs according to their minus domination numbers. For each integer n we determine the smallest order of a connected graph with minus domination number equal to n. Properties of the minus domination number of a graph are presented and a number of open questions are raised. @ 1999 Elsevier Science B.V. All rights reserved