Transactions on Combinatorics (Mar 2019)
A generalization of global dominating function
Abstract
Let $G$ be a graph. A function $f : V (G) \longrightarrow \{0,1\}$, satisfying the condition that every vertex $u$ with $f(u) = 0$ is adjacent with at least one vertex $v$ such that $f(v) = 1$, 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 $s\geq 2$ and $K_n$\ is the complete graph on $V(G)$. A function $ f:V(G)\longrightarrow \{ 0,1\} $ on $G$ is $s$-dominating function $(s-DF)$, if there exists some factorization $\{G_1,\ldots,G_s \}$ of $K_n$, such that $G_1=G$ \ and $f$\ is dominating function of each $G_i$.
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