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

TL;DR

The paper investigates Roman $\{2\}$-domination on graphs with a limited number of induced $P_4$ paths, leveraging modular decomposition and a detailed study of how $\gamma_{\{R2\}}(G)$ behaves under key graph-operations to enable linear-time algorithms. It provides closed-form expressions and structural characterizations for several graph classes (spiders, quasi-spiders, well-labelled spiders, $ZOO$-graphs, and split-$\{H_1,H_2,\overline{H_1},\overline{H_2}\}$-free graphs) and shows how these results extend to broad families through join and union formulas. The work then combines these insights to derive linear-time algorithms for $\gamma_{\{R2\}}(G)$ on cographs, $P_4$-sparse, $P_4$-tidy, and partner-limited graphs, while also establishing NP-completeness on $P_4$-laden graphs to delineate the tractable boundary. Overall, the results deliver practical, structure-aware procedures for computing Roman $\{2\}$-domination in important graph classes with limited $P_4$-patterns, leveraging modular decomposition and clique-width-boundedness for efficiency.

Abstract

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.

Figures (3)

• Figure 1: Graphs $J$, $K$, $L$ and $Q$
• Figure 2: Graphs $M_i$ in ZOO, for $i \in [1, 11]$
• Figure 3: Graphs $H_1$, $H_2$ and their complements

Theorems & Definitions (31)

• Theorem 2.1
• proof
• Proposition 2.2
• proof
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5