Doost Ali Mojdeh

For a graph $G = (V, E)$ with vertex set $V=V(G)$ and edge set $E=E(G)$, a Roman $\{3\}$-dominating function (R$\{3\}$-DF) is a function $f : V(G) \rightarrow \{0, 1, 2, 3\}$ 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 $v\in V(G)$. 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 $\gamma_{\{R3\}}(G)$ \cite{mv}. Let $G$ be a graph with no isolated vertices. The total Roman $\{3\}$-dominating function on $G$ is an R$\{3\}$-DF $f$ on $G$ with the additional property that every vertex $v\in V$ with $f(v)\ne 0$ has a neighbor $w$ with $f(w)\ne 0$. The minimum weight of a total Roman $\{3\}$-dominating function on $G$, is called the total Roman $\{3\}$-domination number denoted by $\gamma_{t\{R3\}}(G)$. We initiate the study of total Roman $\{3\}$-domination and show its relationship to other domination parameters. We present an upper bound on the total 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 investigate the complexity of total Roman $\{3\}$-domination for bipartite graphs.