unicyclic graphs with strong equality between the 2-rainbow domination and independent 2-rainbow domination numbers

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 ∈ v with f (v) = ∅ the condition ∪u∈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 ω(f ) = ∑v ∈v |f (v)|. the 2-rainbow domination number γr2(g) (respectively, the independent 2-rainbow domination number ir2(g) ) is the minimum weight of a 2rdf (respectively, i2rdf) on g. we say that γr2(g) is strongly equal to ir2(g) and denote by γr2(g) ≡ ir2(g), if every 2rdf on g of minimum weight is an i2rdf. in this paper we characterize all unicyclic graphs g with γr2(g) ≡ ir2(g).