A sharp upper bound on the independent 2-rainbow domination in graphs with minimum degree at least two

Abstract

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 $i_{r2}(G)\leq \frac{4n}{5}$.

Keywords

• independent $k$-rainbow dominating function
• independent $k$-rainbow domination number