Abstract
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).
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