Azarbaijan Shahid Madani UniversityCommunications in Combinatorics and Optimization2538-21284220191201On trees with equal Roman domination and outer-independent Roman domination numbers1851991386510.22049/cco.2019.26319.1095ENSeyed Mahmoud SheikholeslamiAzarbaijan Shahid Madani University0000-0003-2298-4744Sakineh Nazari-MoghaddamAzarbaijan Shahid Madani UniversityJournal Article20180802A 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 $\gamma_{R}(T)=\gamma_{oiR}(T)$. https://comb-opt.azaruniv.ac.ir/article_13865_778d0f97a1447e3fa6dcc653002a9d16.pdf