Lower bounds on the minus domination and k-subdomination numbers

A three-valued function f defined on the vertex set 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 all vertices adjacent to v. The weight of a minus function is f(V) = ∑_v∈Vf(v). The minus domination number of a graph G, denoted by γ^-(G), equals the minimum weight of a minus dominating function of G. In this paper, sharp lower bounds on minus domination of a bipartite graph are given. Thus, we prove a conjecture proposed by Dunbar et al. (Discrete Math. 199 (1999) 35), and we give a lower bound on γ_ks(G) of a graph G.