Perfect Italian domination in trees

Abstract

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 of f 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 γIp(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 γIp(G)≤45n, and for each positive integer n≡0(mod5) there exists a tree of order n for which equality holds in the bound.