Averaging 2-rainbow domination and Roman domination

For a graph GG, let γr2(G)γr2(G) and γR(G)γR(G) denote the 2-rainbow domination number and the Roman domination number, respectively. Fujita and Furuya (2013) proved γr2(G)+γR(G)≤64n(G) for a connected graph GG of order n(G)n(G) at least 3. Furthermore, they conjectured γr2(G)+γR(G)≤43n(G) for a connected graph GG of minimum degree at least 2 that is distinct from C5C5. We characterize all extremal graphs for their inequality and prove their conjecture.