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).