Abstract
Let [Formula: see text] be a graph with vertex set [Formula: see text]. A double Roman dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the conditions that if [Formula: see text], then vertex [Formula: see text] must have at least two neighbors in [Formula: see text] or one neighbor in [Formula: see text]; if [Formula: see text], then vertex [Formula: see text] must have at least one neighbor in [Formula: see text]. The weight of a double Roman dominating function [Formula: see text] is the sum [Formula: see text], and the double Roman domination number [Formula: see text] is the minimum weight of a double Roman dominating function on [Formula: see text]. A double Italian dominating function on a graph [Formula: see text] is a function [Formula: see text] satisfying the condition that for every vertex [Formula: see text], if [Formula: see text], then [Formula: see text]. The double Roman domination number [Formula: see text] is the minimum weight of a double Italian dominating function on [Formula: see text]. Mojdeh and Volkmann [D.A. Mojdeh and L. Volkmann, Roman 3-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555–564] proved that [Formula: see text] for any tree [Formula: see text]. However, we find that there is a minor issue in the proof. In this paper, we first prove that [Formula: see text]. Subsequently, we present a sharp bound on the double Italian domination number of any non-trivial tree [Formula: see text], and characterize the trees attaining this bound.