Given a graph \(G=(V,E)\) and a function \(b: V\rightarrow \{0, 1, 2\}\) . If b satisfies \(\sum _{z\in N_G(w)}b(z)\ge 2\) for every vertex w with \(b(w)=0\) , where \(N_G(w)\) is the neighborhood of w in G, then b is called a Roman {2}-dominating function of G. The Roman {2}-domination number \(\gamma _{\{R2\}}(G)\) of G is the minimum value \(\sum _{w\in V}b(w)\) among all Roman {2}-dominating functions b of G. A set \(S\subseteq V\) is a 2-dominating set of G if \(|N_G(w)\cap S|\ge 2\) for each \(w\in V\backslash S\) . The minimum cardinality among all 2-dominating sets of G is called the 2-domination number \(\gamma _2(G)\) of G. For any graph G, Chellali et al. (2016) showed that \(\gamma _{\{R2\}}(G)\le \gamma _{2}(G)\) . In this article, firstly, we provide a characterization for the trees T with \(\gamma _{\{R2\}}(T)=\gamma _{2}(T)\) . Secondly, for a tree T, we provide a lower bound of \(\gamma _{\{R2\}}(T)\) relying on \(\gamma _{2}(T)\) and the number of leaves l(T) in T: \(\gamma _{\{R2\}}(T)\ge \gamma _{2}(T)-l(T)+2\) , and a characterization for the trees T with \(\gamma _{\{R2\}}(T)=\gamma _{2}(T)-l(T)+2\) . Finally, we prove that it is NP-hard to determine whether \(\gamma _{\{R2\}}(G)\) and \(\gamma _2(G)\) are equal for a given bipartite graph G.
