Abstract
A 2-rainbow dominating function (2RDF) on a graph $$G=(V, E)$$ is a function f from the vertex set V to the set of all subsets of the set $$\{1,2\}$$ such that for any vertex $$v\in V$$ with $$f(v)=\emptyset $$ the condition $$\bigcup _{u\in N(v)}f(u)=\{1,2\}$$ is fulfilled. A 2RDF f is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value $$\omega (f)=\sum _{v\in V}|f (v)|$$ . The 2-rainbow domination number $$\gamma _{r2}(G)$$ (respectively, the independent 2-rainbow domination number $$i_{r2}(G)$$ ) is the minimum weight of a 2RDF (respectively, I2RDF) on G. We say that $$\gamma _{r2}(G)$$ is strongly equal to $$i_{r2}(G)$$ and denote by $$\gamma _{r2}(G)\equiv i_{r2}(G)$$ , if every 2RDF on G of minimum weight is an I2RDF. In this paper, we provide a constructive characterization of trees T with $$\gamma _{r2}(T)\equiv i_{r2}(T)$$ .
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