Relating $2$-rainbow domination to weak Roman domination

TL;DR

This paper establishes an inequality relating 2-rainbow domination and weak Roman domination numbers in graphs, characterizes extremal graphs for this inequality, and discusses the computational complexity of recognizing certain extremal graphs.

Contribution

It proves the inequality -rainbow domination number   weak Roman domination number for all graphs and characterizes extremal graphs that are  and -e-free.

Findings

Proves -rainbow domination   weak Roman domination inequality.

Characterizes extremal graphs for the inequality that are  and -e-free.

Shows recognition of certain extremal graphs is NP-hard.

Abstract

Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27-32) we prove $\gamma_{r2}(G)\leq 2\gamma_r(G)$ for every graph $G$, where $\gamma_{r2}(G)$ and $\gamma_r(G)$ denote the $2$-rainbow domination number and the weak Roman domination number of $G$, respectively. We characterize the extremal graphs for this inequality that are $\{ K_4,K_4-e\}$-free, and show that the recognition of the $K_5$-free extremal graphs is NP-hard.