Minus domination in graphs

Abstract

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 |:V → {−1,0,1}. Such a function 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 ν ϵ V, |(N[ν])⩾ 1, where N[ν] consists of ν and every vertex adjacent to ν. The weight of a minus dominating function is |(V) = Σ|(ν), over all vertices ν ϵ V. The minus domination number of a graph G, denoted γ⁻(G), equals the minimum weight of a minus dominating function of G. For every graph G, γ⁻(G)⩽γ(G) where γ(G) denotes the domination number of G. We show that if T is a tree of order n⩾4, then γ(T)−γ⁻(T)⩽(n−4)/5 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.