Doost Ali Mojdeh

For a graph $G = (V, E)$ with $V=V(G)$ and $E=E(G)$, a Roman $\{3\}$-dominating function is a function $f : V \rightarrow \{0, 1, 2, 3\}$ having the property that $\sum_{u\in N_G(v)} f(u)\ge 3$, if $f (v) = 0$, and $\sum_{u\in N_G(v)} f(u)\ge 2$, if $f (v) = 1$ for any vertex $v\in G$. The weight of a Roman $\{3\}$-dominating function $f$ is the sum $f (V) =\sum_{v\in V(G)} f(v)$ and the minimum weight of a Roman $\{3\}$-dominating function on $G$ is the Roman $\{3\}$-domination number of $G$, denoted by $\gamma_{\{R3\}}(G)$. We initiate the study of Roman $\{3\}$-domination and show its relationship to domination, Roman domination, Roman $\{2\}$-domination (Italian domination) and double Roman domination. Finally, we present an upper bound on the Roman $\{3\}$-domination number of a connected graph $G$ in terms of the order of $G$ and characterize the graphs attaining this bound. Finally, we show that associated decision problem for Roman $\{3\}$-domination is $NP$-complete, even for bipartite graphs.