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Abstract
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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 f1; 2g such that for any vertex v 2 V with
f(v) = ∅ the condition ∪u2N(v) f(u) = f1; 2g is ful lled. A 2RDF f is independent (I2RDF) if no two
vertices assigned nonempty sets are adjacent. The weight of a 2RDF f is the value !(f) = Σv2V jf(v)j.
The 2-rainbow domination number r2(G) (respectively, the independent 2-rainbow domination number
ir2(G) ) is the minimum weight of a 2RDF (respectively, I2RDF) on G. We say that r2(G) is strongly
equal to ir2(G) and denote by r2(G) ir2(G), if every 2RDF on G of minimum weight is an I2RDF.
In this paper we characterize all unicyclic graphs G with r2(G) ir2(G).
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