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 v ∈ V, f(N[v]) ⩾ 1, where N[v] consists of v and every vertex adjacent to v. The weight of a minus dominating function is f(V) = Σ f(v), over all vertices v ∈ 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.