Independent 2-rainbow domination in graphs

A \emph{2-rainbow dominating function} 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 \( g(v) = \emptyset \) 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 \emph{2-rainbow domination number} \( \gamma_{r2}(G) \). A 2-rainbow dominating function \( g \) of a graph \( G \) is independent if no two vertices assigned non-empty sets are adjacent. The \emph{independent 2-rainbow domination number} \( i_{r2}(G) \) 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.