Abstract

Let $G$= $(V,E)$ be a graph. A Roman domination function (RDF) of $G$ is a Function $f:V\rightarrow\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ is adjacent to a vertex $u$ with $f(u)=2$. The weight of $f$, denoted by $w(f)$, is defined as ${\sum_{v \in V}f(v) }.$ The Roman domination number of $G$, denoted by $\gamma_ R(G)$, is defined as \begin{center} $\gamma_ R(G)= min\{w(f): f$ is a RDF of $G$\}. \end{center} In this talk, we introduce three different generalizations of RDF of $G$ with emphasis on the recent one given by Gunawan and Koh. The lower bounds for the corresponding number of this generalization in terms of the diameter and radius of $G$ will be presented..



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