Relating 2-rainbow Domination to Weak Roman Domination

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.