دوستعلی مژده

Let $G=(V,E)$ be a graph. A Roman $\{2\}$-dominating function $f : V \rightarrow \{0, 1, 2\}$ such that for every vertex $v \in V$, with $f (v) = 0,\ f (N(v)) \ge 2$, Chellali et al \cite{chhm}.\\ A double Roman Dominating function on a graph $G$ is a function $ f:V\rightarrow \{0,1,2,3\}$ such that the following conditions are met:\\ (a) if $f(v)=0$, then vertex $v$ must have at least two neighbors $x,y$ with $f(x)=f(y)=2$ or one neighbor $z$ with $f(z)=3$.\\ (b) if $f(v)=1$ , then vertex $v$ must have at least one neighbor $x$ with $f(x)\ge 2$.\\ The weight of a double Roman dominating function is the sum $w_f=\sum_{v\in V(G)}{f(v)}$, Beeler et al \cite{bhh}.\\ Here these two concepts are extended as follows:\\ \textbf{Definition.} For a graph $G$, a Roman $\{3\}$-dominating function is a function $f : V \rightarrow \{0, 1, 2, 3\}$ having the property that for every vertex $u \in V$, if $f(u)\in \{0,1\}$, then $f (N[u]) \ge 3$. The weight of a Roman $\{3\}$-dominating function