Minus domination in regular graphs

A three-valued function f defined on the vertices of a graph G = (V,E), f : V → {-1,0,1}, is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every υ ∈ V, f(N[υ]) ≥ 1, where N[υ] consists of υ and every vertex adjacent to υ. The weight of a minus dominating function is f(V) = Σ f(υ), over all vertices υ ∈ V. The minus domination number of a graph G, denoted γ^-(G), equals the minimum weight of a minus dominating function of G. In this note, we establish a sharp lower bound on γ^-(G) for regular graphs G.