Summary: For a graph \(G = (V, E)\), a Roman \(\{2\}\)-dominating function (R2DF) \(f : V \rightarrow \{0, 1, 2\}\) has the property that for every vertex \(v \in V\) with \(f(v) = 0\), either there exists a neighbor \(u \in N(v)\), with \(f(u) = 2\), or at least two neighbors \(x, y \in N(v)\) having \(f(x) = f(y) = 1\). The weight of an R2DF \(f\) is the sum \(f (V) = \sum_{v \in V} f(v)\), and the minimum weight of an R2DF on \(G\) is the Roman \(\{2\}\)-domination number \(\gamma_{\{R2\}}(G)\). An R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman \(\{2\}\)-domination number \(i_{\{R2\}}(G)\) is the minimum weight of an independent Roman \(\{2\}\)-dominating function on \(G\). In this paper, we show that the decision problem associated with \(\gamma_{\{R2\}}(G)\) is NP-complete even when restricted to split graphs. We design a linear time algorithm for computing the value of \(i_{\{R2\}}(T)\) in any tree \(T\), which answers an open problem raised by A. Rahmouni and M. Chellali [Discrete Appl. Math. 236, 408–414 (2018; Zbl 1377.05145)]. Moreover, we present a linear time algorithm for computing the value of \(\gamma_{\{R2\}}(G)\) in any block graph \(G\), which is a generalization of trees.