Doost Ali Mojdeh

For a graph $G = (V, E)$ with $V=V(G)$ and $E=E(G)$, a perfect Roman $\{3\}$-dominating function is a function $f : V \rightarrow \{0, 1, 2, 3\}$ having the property that $3\leq\sum_{u\in N_G[v]} f(u)\leq 4$, if $f (v) \in\{ 0,1\}$ for any vertex $v\in G$. The weight of a perfect Roman $\{3\}$-dominating function $f$ is the sum $f (V) =\sum_{v\in V(G)} f(v)$ and the minimum weight of a perfect Roman $\{3\}$-dominating function on $G$ is the perfect Roman $\{3\}$-domination number of $G$, denoted by $\gamma_{\{PR3\}}(G)$. In this manuscript we study the perfect Roman $\{3\}$-domination of some graphs.