Addressing a problem posed by Chellali, Haynes, and Hedetniemi (Discrete Appl. Math. 178 (2014) 27–32), we prove for every graph , where and denote the 2-rainbow domination number and the weak Roman domination number of , respectively. We characterize the extremal graphs for this inequality that are -free, and show that the recognition of the -free extremal graphs is NP-hard.
Keywords: 2-rainbow domination; Roman domination; weak Roman domination.