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

A 2-emph{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 $vin V$ with $f(v)=emptyset$ the condition $bigcup_{uin N(v)}f(u)={1,2}$ is fulfilled. A 2RDF $f$ is independent (I2RDF) if no two vertices assigned nonempty sets are adjacent. The emph{weight} of a 2RDF $f$ is the value $omega(f)=sum_{vin V}|f (v)|$. The 2-emph{rainbow domination number} $gamma_{r2}(G)$ (respectively, the emph{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 characterize all unicyclic graphs $G$ with $gamma_{r2}(G)equiv i_{r2}(G)$.