# Total Roman {3}-domination in Graphs

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Institute of Computing Science and Technology, Guangzhou University, Guangzhou 510006, China

^{2}

Department of Mathematics, University of Mazandaran, Babolsar 47416-95447, Iran

^{3}

Lehrstuhl II für Mathematik, RWTH Aachen University, 52056 Aachen, Germany

^{*}

Author to whom correspondence should be addressed.

Received: 4 January 2020 / Revised: 2 February 2020 / Accepted: 3 February 2020 / Published: 9 February 2020

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

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**MDPI and ACS Style**

Shao, Z.; Mojdeh, D.A.; Volkmann, L. Total Roman {3}-domination in Graphs. *Symmetry* **2020**, *12*, 268.

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