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.