Global minus domination in graphs

A function $f:V(G)rightarrow {-1,0,1}$ is a {em minus dominating function} if for every vertex $vin V(G)$, $sum_{uin N[v]}f(u)ge 1$. A minus dominating function $f$ of $G$ is called a {em global minus dominating function} if $f$ is also a minus dominating function of the complement $overline{G}$ of $G$. The {em global minus domination number} $gamma_{g}^-(G)$ of $G$ is defined as $gamma_{g}^-(G)=min{sum_{vin V(G)} f(v)mid f mbox{ is a global minus dominating function of } G}$. In this paper we initiate the study of global minus domination number in graphs and we establish lower and upper bounds for the global minus domination number.