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 Trees with equal strong Roman domination number and Roman domination number Bull. Korean Math. Soc. 2019 Vol. 56, No. 1, 31-44 https://doi.org/10.4134/BKMS.b180055Published online 2019 Jan 31 Xue-Gang Chen, Moo Young Sohn North China Electric Power University; Changwon National University Abstract : A graph theoretical model called Roman domination in \linebreak graphs originates from the historical background that any undefended place (with no legions) of the Roman Empire must be protected by a stronger neighbor place (having two legions). It is applicable to military and commercial decision-making problems. A Roman dominating function for a graph $G = (V,E)$ is a function $f : V \to \{0, 1, 2 \}$ such that every vertex $v$ with $f(v) = 0$ has at least a neighbor $w$ in $G$ for which $f(w) = 2$. The Roman domination number of a graph is the minimum weight $\sum_{v \in V} f(v)$ of a Roman dominating function. In order to deal a problem of a Roman domination-type defensive strategy under multiple simultaneous attacks, $\acute{A}$lvarez-Ruiz et al.~\cite{Ruiz} initiated the study of a new parameter related to Roman dominating function, which is called strong Roman domination. $\acute{A}$lvarez-Ruiz et al. posed the following problem: Characterize the graphs $G$ with equal strong Roman domination number and Roman domination number. In this paper, we construct a family of trees. We prove that for a tree, its strong Roman dominance number and Roman dominance number are equal if and only if the tree belongs to this family of trees. Keywords : Roman domination number, strong Roman domination number, tree MSC numbers : 05C69, 05C38 Full-Text :