دوستعلی مژده

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 $\gamma_{\{R3\}}(G)$. The total Roman $\{3\}$-dominating function on a graph $G$ with no isolated vertices is a TR$\{3\}$-DF $f$ on $G$ with the additional property that the subgraph of $G$ induced by the set $\{v\in V:f(v)\ne0\}$ has no isolated vertices. The total Roman $\{3\}$-domination number $\gamma_{tR\{3\}}(G)$ is the minimum weight of a total Roman $\{3\}$-dominating function on $G$. We initiate the study of total Roman $\{3\}$-domination and show its relationship to other domination parameters. Finally, 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.