# A characterization relating domination, semitotal domination and total Roman domination in trees

Document Type : Original paper

Authors

1 Universitat Rovira i Virgili, Tarragona, Spain

2 Departamento de Matem&aacute;tica, Universidad de Oriente, Cuba

Abstract

A total Roman dominating function on a graph $G$ is a function $f: V(G) \rightarrow \{0,1,2\}$ such that for every vertex $v\in V(G)$ with $f(v)=0$ there exists a vertex $u\in V(G)$ adjacent to $v$ with $f(u)=2$, and the subgraph induced by the set $\{x\in V(G): f(x)\geq 1\}$ has no isolated vertices. The total Roman domination number of $G$, denoted $\gamma_{tR}(G)$, is the minimum weight $\omega(f)=\sum_{v\in V(G)}f(v)$ among all total Roman dominating functions $f$ on $G$. It is known that $\gamma_{tR}(G)\geq \gamma_{t2}(G)+\gamma(G)$ for any graph $G$ with neither isolated vertex nor components isomorphic to $K_2$, where $\gamma_{t2}(G)$ and $\gamma(G)$ represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above.

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