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)=∅ we have ⋃u∈N(v)g(u)={1,2}. The weight of a 2-rainbow dominating function g is the value w(g)=∑v∈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 γr2(G) (respectively, the independent 2-rainbow domination number ir2(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≥3, with ℓ leaves and s support vertices, ir2(T)≤(14n+ℓ+s)/20, thus improving the bound given in [Independent 2-rainbow domination in trees, Asian-Eur. J. Math. 8 (2015) 1550035] under certain conditions.