Roman $\{2\}$-domination on Graphs with "few" 4-paths

Given a graph $G$ with vertex set $V$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-dominating function} (or \emph{italian dominating function}) of $G$ if for every vertex $v\in V$ with $f(v) =0$, either there exists a vertex $u$ adjacent to $v$ with $f(u) = 2$, or two distinct vertices $x,\; y$ both adjacent to $v$ with $f(x)=f(y)=1$. The decision problem associated with Roman $\{2\}$-domination is NP-complete even for bipartite graphs (Chellali et al., 2016). In this work we initiate the study of Roman $\{2\}$-domination on graph classes with a limited number of 4-paths. We base our study on a modular decomposition analysis. In particular, we study Roman $\{2\}$-domination under some operations in graphs such as join, union, complementation, addition of pendant vertices and addition of twin vertices. We then obtain the Roman $\{2\}$-domination number of spiders, well-labelled spiders and certain prime split graphs that are crucial in the modular decomposition of partner-limited graphs. In all, we provide linear-time algorithms to compute the Roman $\{2\}$-domination number of cographs, $P_4$-sparse graphs, $P_4$-tidy graphs and partner-limited graphs. Finally, we derive the NP-completeness of Roman $\{2\}$-domination on $P_4$-laden graphs.