A new upper bound on the independent $2$-rainbow domination number in trees

A $2$-rainbow dominating function on a graph $G$ is a function $g$ that assigns to each vertex a set of colors chosen from the subsets of ${1, 2}$ so that for each vertex with $g(v) =emptyset$ we have $bigcup_{uin N(v)} g(u) = {1, 2}$. The weight of a $2$-rainbow dominating function $g$ is the value $w(g) = sum_{vin V(G)} |f(v)|$. A $2$-rainbow dominating function $g$ is an independent $2$-rainbow dominating function if no pair of vertices assigned nonempty sets are adjacent. The $2$-rainbow domination number $gamma_{r2}(G)$ (respectively, the independent $2$-rainbow domination number $i_{r2}(G)$) is the minimum weight of a $2$-rainbow dominating function (respectively, independent $2$-rainbow dominating function) on $G$. We prove that for any tree $T$ of order $ngeq 3$, with $ell$ leaves and $s$ support vertices, $i_{r2}(T)leq (14n+ell+s)/20$, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions.