  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.
