A new upper bound for the perfect Italian domination number of a tree

A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γ I (G), is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with `(T ) leaves and s(T ) support vertices, γ I (T ) ≤ 4n−`(T )+2s(T )−1 5 , improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164– 177].