Minus domination number in cubic graph

An upper bound is established on the parameter Γ⁻ (G) for a cubic graphG and two infinite families of 3-connected graphs G_k, G_k* are constructed to show that the bound is sharp and, moreover, the difference Γ⁻(G_k*)-γ_s(G_k*) can be arbitrarily large, where Γ⁻(G_k*) and γ_s(G_k*) are the upper minus domination and signed domination numbers of G_k*, respectively. Thus two open problems are solved.