On trees with equal Roman domination and outer-independent Roman domination number

Abstract

A Roman dominating function (RDF) on a graph $G$ is a function $f : V (G) \to \{0, 1, 2\}$ satisfying the condition that every vertex $u$ for which $f(u) = 0$ is adjacent to at least one vertex $v$ for which $f(v) = 2$. A Roman dominating function $f$ is called an outer-independent Roman dominating function (OIRDF) on $G$ if the set $\{v\in V\mid f(v)=0\}$ is independent. The (outer-independent) Roman domination number $\gamma_{R}(G)$ ($\gamma_{oiR}(G)$) is the minimum weight of an RDF (OIRDF) on $G$. Clearly for any graph $G$, $\gamma_{R}(G)\le \gamma_{oiR}(G)$. In this paper, we provide a constructive characterization of trees $T$ with $\gamma_{R}(T)=\gamma_{oiR}(T)$.

Keywords

• Roman domination
• outer-independent Roman domination
• tree