A generalization of global dominating function

Document Type: Research Paper

Authors

1 Department of‎ ‎Mathematics‎, ‎Shahid Rajaee Teacher Training University‎, ‎P.O‎. ‎Box 16785-163, Tehran‎, ‎Iran

2 Department of math, Shahid Rajaee Teacher Training University, Tehran, Iran

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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