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 Gk, Gk* are constructed to show that the bound is sharp and, moreover, the difference Γ−(Gk*)-γs(Gk*) can be arbitrarily large, where Γ−(Gk*) and γs(Gk*) are the upper minus domination and signed domination numbers of Gk*, respectively. Thus two open problems are solved.