Relating $2$-Rainbow Domination to Roman domination

For a graph $G$, let $γ_R(G)$ and $γ_{r2}(G)$ denote the Roman domination number of $G$ and the $2$-rainbow domination number of $G$, respectively. It is known that $γ_{r2}(G)\leq γ_R(G)\leq \frac{3}{2}γ_{r2}(G)$. 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 $γ_R(G)-γ_{r2}(G)=k$ 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 $γ_R(G)-γ_{r2}(G)=k$ is NP-hard, which implies that there is most likely no good characterization of these graphs. We characterize the graphs $G$ such that $γ_{r2}(H)=γ_R(H)$ for every induced subgraph $H$ of $G$, and collect several properties of the graphs $G$ with $γ_R(G)=\frac{3}{2}γ_{r2}(G)$.

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