Lower bounds on the minus domination and k-subdomination numbers

Abstract

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∈ V f( 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.