The signed star domination numbers of the Cartesian product graphs

Let G   be a graph with vertex set V(G)V(G) and edge set E(G)E(G). A function f:E(G)→{-1,1}f:E(G)→{-1,1} is said to be a signed star dominating function of G   if ∑e∈EG(v)f(e)⩾1 for every v∈V(G)v∈V(G), where EG(v)={uv∈E(G)|u∈V(G)}EG(v)={uv∈E(G)|u∈V(G)}. The minimum of the values of ∑e∈E(G)f(e), taken over all signed star dominating functions f   on GG, is called the signed star domination number of G   and is denoted by γSS(G)γSS(G). In this paper, a sharp upper bound of γSS(G×H)γSS(G×H) is presented.