A generalization of global dominating function

Let $G$ be a graph‎. ‎A function $f‎ : ‎V (G) \longrightarrow \{۰,۱\}$‎, ‎satisfying‎ ‎the condition that every vertex $u$ with $f(u) = ۰$ is adjacent with at‎ ‎least one vertex $v$ such that $f(v) = ۱$‎, ‎is called a dominating function $(DF)$‎. ‎The weight of $f$ is defined as $wet(f)=\Sigma_{v \in V(G)} f(v)$‎. ‎The minimum weight of a dominating function of $G$‎ ‎is denoted by‎ ‎$\gamma (G)$‎, ‎and is called the domination number of $G$‎. ‎A dominating‎ ‎function $f$ is called a global dominating function $(GDF)$ if $f$ is‎ ‎also a $DF$ of $\overline{G}$‎. ‎The minimum weight of a global dominating function is denoted by‎ ‎$\gamma_{g}(G)$ and is called global domination number of $G$‎. ‎In this paper we introduce a generalization of global dominating function‎. ‎Suppose $G$ is a graph and $s\geq ۲$ and $K_n$\ is the complete graph on $V(G)$‎. ‎A function $ f:V(G)\longrightarrow \{ ۰,۱\} $ on $G$ is $s$-dominating function $(s-DF)$‎, ‎if there exists some factorization $\{G_۱,\ldots,G_s \}$ of $K_n$‎, ‎such that $G_۱=G$ \ and $f$\ is dominating function of each $G_i$‎.

Keywords:

‎‎dominating function‎ , ‎global dominating function‎ , ‎$s$-dominating function‎ , ‎$gamma-$function‎ , ‎$gamma_s-$function