Perfect Italian domination in trees

A perfect Italian dominating function 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 off assigned to the neighbors of u is exactly two. The weight of a perfect Italian dominating function is the sum of the weights of the vertices. The perfect Italian domination number of G, denoted gamma(p)(I)(G), is the minimum weight of a perfect Italian dominating function of G. We show that if G is a tree on n >= 3 vertices, then gamma(p)(I)(G) <= 4/5n, and for each positive integer n (math) 0 (mod 5) there exists a tree of order n for which equality holds in the bound. (C) 2019 Elsevier B.V. All rights reserved.