| Summary: | An independent 2-rainbow dominating function (I$2$-RDF) on a graph $G$ is a
function $f$ from the vertex set $V(G)$ to the set of all subsets of the set
$\{1,2\}$ such that $\{x\in V\mid f(x)\neq \emptyset \}$ is an independent
set of $G$ and for any vertex $v\in V(G)$ with $f(v)=\emptyset $ we have $%
\bigcup_{u\in N(v)}f(u)=\{1,2\}$. The \emph{weight} of an I$2$-RDF $f$ is the
value $\omega (f)=\sum_{v\in V}|f(v)|$, and the independent $2$-rainbow
domination number $i_{r2}(G)$ is the minimum weight of an I$2$-RDF on $G$. In
this paper, we prove that if $G$ is a graph of order $n\geq 3$ with minimum
degree at least two such that the set of vertices of degree at least $3$ is
independent, then $i_{r2}(G)\leq \frac{4n}{5}$.
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