Upper minus domination in regular graphs Original Research Article

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)=∑v∈Vf(v). The upper minus domination number of a graph G, denoted Γ−(G), equals the maximum weight of a minimal minus dominating function of G. In this paper, sharp upper bounds on Γ− of regular graphs are found. Thus, we answer an open problem proposed by Henning and Slater (Discrete Math. 158 (1996) 87–98).