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Abstract
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For a graph G, let γ r (G), γ R (G) and γ r2 (G) denote the weak Roman
domination number, the Roman domination number and the 2-rainbow
domination number, respectively. It is well-known that for every graph G,
γ r (G) ≤ γ r2 (G) ≤ γ R (G). In this paper, we characterize all trees T with
γ r (T) = γ r2 (T) or γ r (T) = γ R (T) answering two open problems posed
by Chellali, Haynes and Hedetniemi [Discrete Appl. Math. 178 (2014),
27–32
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