New complexity results on Roman {2}-domination

The study of a variant of Roman domination was initiated by Chellali et al. (2016).  Given a graph $G$ with vertex set $V$, a Roman $\{2\}$-dominating function $f : V \rightarrow \{0, 1, 2\}$ has the property that for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex $u$ adjacent to $v$ with $f(u) = 2$, or at least two vertices $x,\; y$  adjacent  to $v$ with $f(x)=f(y)=1$. The weight of a Roman $\{2\}$-dominating function is the value $f(V) = \sum_{v\in V} f(v)$. The minimum weight of a Roman $\{2\}$-dominating function is called the Roman $\{2\}$-domination number and is denoted by $\gamma_{\{R2\}}(G)$.  In this work we find several NP-complete instances of the Roman  $\{2\}$-domination problem: chordal graphs, bipartite planar graphs, chordal bipartite graphs, bipartite with maximum degree 3 graphs, among others. A result by Chellali et al. (2016) shows that $\gamma_{\{R2\}}(G)$ and the 2-rainbow domination number of $G$ coincide when $G$ is a tree, and thus, the linear time algorithm for $k$-rainbow domination due to Bresar et al. (2008) can be followed to compute $\gamma_{\{R2\}}(G)$. In this work we develop an efficient algorithm that is independent of $k$-rainbow domination and computes the Roman $\{2\}$-domination number on a subclass of trees called caterpillars.