Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 8 1 2023 03 01 A new upper bound on the independent $2$-rainbow domination number in trees 261 270 14355 10.22049/cco.2022.27641.1305 EN Nader Jafari Rad Shahed University Elham Gholami Islamic Azad University A Tehranian ‎Islamic Azad University Hamid Rasouli Islamic Azad University, Journal Article 2021 11 18 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 $\bigcup_{u\in N(v)} g(u) = \{1, 2\}$. The weight of a $2$-rainbow dominating function $g$ is the value $w(g) = \sum_{v\in V(G)} |f(v)|$. 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 $n\geq 3$, 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. https://comb-opt.azaruniv.ac.ir/article_14355_4f1fe4a7a66211a655541d57e894afed.pdf