Bounds on the 2-rainbow domination number of graphs

A {\it 2-rainbow domination function} of a graph $G$ is a function $f$ that assigns to each vertex a set of colors chosen from the set $\{1,2\}$, such that for any $v\in V(G)$, $f(v)=\emptyset$ implies $\bigcup_{u\in N(v)}f(u)=\{1,2\}$. The {\it 2-rainbow domination number $γ_{r2}(G)$} of a graph $G$ is the minimum $w(f)=Σ_{v\in V}|f(v)|$ over all such functions $f$. Let $G$ be a connected graph of order $|V(G)|=n\geq 3$. We prove that $γ_{r2}(G)\leq 3n/4$ and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of $γ_{r2}(G)$ in terms of diameter are also given.