On $\{2\}$-Roman graph recognition of Partner Limited graphs

Abstract

Given a graph $G=(V,E)$, $f : V \rightarrow \{0, 1, 2\}$ is a \emph{Roman $\{2\}$-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 $f(x)=f(y)=1$ (Chellali et al. 2016). Every graph $G$ satisfies $γ_{\{R2\}}(G) \leq 2γ(G)$, where $γ_{\{R2\}}(G)$ denotes the minimum weight of a $\{2\}$-Roman dominating function of $G$ and $γ(G)$ is the domination number of $G$. \emph{$\{2\}$-Roman graphs} are those for which the equality is reached (Klostermeyer et al. 2019). A characterization of $\{2\}$-Roman trees was given by Henning et al. in 2017. In 2025, Ferrari et al. characterized the $\{2\}$-Roman property by the existence of a minimum $\{2\}$-Roman dominating function of $G$ that assumes only $0, 2$-values. Afterwards in 2025, Bešter Štorgel et al. introduced the problem of recognizing $\{2\}$-Roman graphs, proved polinomiality for middle graphs, and characterized \hbox{$\{2\}$-Roman} split graphs that can be decomposed with respect to the split join operation into two smaller split graphs. Recognition complexity is still open for general graphs. In this paper we study the \hbox{$\{2\}$-Roman} property on graphs that can be decomposed into two smaller graphs with respect to the join and union operations, allowing to completely characterize the $\{2\}$-Roman property. The 4-path is the non trivial connected non $\{2\}$-Roman graph with the fewest number of vertices and edges. We classify the $\{2\}$-Roman property within specific families of non decomposable graphs with a limited number of 4-paths which are present in the decomposition of partner limited graphs; these are well-labelled spiders, the graphs in ZOO and some special split graphs.