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 $$ f\left( V \right) = \sum _{\upsilon \in V} f\left( \upsilon \right) $$. 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 J. Dunbar etc.(Discrete Math. 199(1999) 35-47), and we give a lower bound on γ_ka(G) of a graph G.