  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 $\gamma_R(G)=\frac{3}{2}\gamma_{r2}(G)$.
