On the Minus Domination Number of Graphs

Abstract

Let G = (V, E) be a simple graph. A 3-valued function is said to be a minus dominating function if for every vertex where N[v] is the closed neighborhood of v. The weight of a minus dominating function f on G is The minus domination number of a graph G, denoted by γ⁻(G), equals the minimum weight of a minus dominating function on G. In this paper, the following two results are obtained.

(1) If G is a bipartite graph of order n, then

(2) For any negative integer k and any positive integer m ⩾ 3, there exists a graph G with girth m such that γ⁻(G) ≤ k. Therefore, two open problems about minus domination number are solved.