Relating $2$-Rainbow Domination to Roman domination

TL;DR

This paper investigates the computational complexity of recognizing graphs with specific differences between Roman and 2-rainbow domination numbers, showing NP-hardness for fixed differences and characterizing certain graph classes.

Contribution

It proves NP-hardness for recognizing connected K4-free graphs with fixed Roman-rainbow domination difference and characterizes graphs where these domination parameters coincide or reach the upper bound.

Findings

Recognition problem is NP-hard for fixed k.

Graphs with equal domination parameters are characterized.

Properties of graphs with maximum ratio are collected.

Abstract

For a graph $G$, let $\gamma_R(G)$ and $\gamma_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $\gamma_{r2}(G)\leq \gamma_R(G)\leq \frac{3}{2}\gamma_{r2}(G)$. Fujita and Furuya (Difference between 2-rainbow domination and Roman domination in graphs, Discrete Applied Mathematics 161 (2013) 806-812) present some kind of characterization of the graphs $G$ for which $\gamma_R(G)-\gamma_{r2}(G)=k$ for some integer $k$. Unfortunately, their result does not lead to an algorithm that allows to recognize these graphs efficiently. We show that for every fixed non-negative integer $k$, the recognition of the connected $K_4$-free graphs $G$ with $\gamma_R(G)-\gamma_{r2}(G)=k$ is NP-hard, which implies that there is most likely no good characterization of these graphs. We characterize the graphs $G$ such that $\gamma_{r2}(H)=\gamma_R(H)$ for every induced subgraph $H$ of $G$, and collect several properties of the graphs $G$ with $\gamma_R(G)=\frac{3}{2}\gamma_{r2}(G)$.