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 afunction $f$ from the vertex set $v$ to the set of all subsets of the set${1,2}$ such that for any vertex $vin v$ with $f(v)=emptyset$ thecondition $bigcup_{uin n(v)}f(u)={1,2}$ is fulfilled. a 2rdf $f$ isindependent (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, theemph{independent $2$-rainbow domination number } $i_{r2}(g)$ ) is the minimumweight of a 2rdf (respectively, i2rdf) on $g$. we say that $gamma_{r2}(g)$ isstrongly 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 wecharacterize all unicyclic graphs $g$ with $gamma_{r2}(g)equiv i_{r2}(g)$.