Independent $2$-rainbow Domination in Graphs

Abstract

A $2-rainbowdominatingfunction$ of a graph $G$ is a function $g$ that assigns to each vertex a set of colors chosen from the set $\{1,2\}$ so that for each vertex $v$ with $g(v)=\mathrm{\varnothing }$ we have ${\cup }_{u\in N(v)}g(u)=\{1,2\}$. The minimum of $g(V(G))=\sum _{v\in V(G)}|g(v)|$ over all such functions is called the $2-rainbowdominationnumber$ ${\gamma }_{r2}(G)$. A $2$-rainbow dominating function $g$ of a graph $G$ is independent if no two vertices assigned non-empty sets are adjacent. The $independent2-rainbowdominationnumber$ $i_{r2}(G)$ is the minimum weight of an independent $2$-rainbow dominating function of $G$. In this paper, we study independent $2$-rainbow domination in graphs. We present some bounds and relations with other domination parameters.