Double Italian domination in trees

Abstract

Let $G$ be a graph with vertex set $V=V(G)$. A double Roman dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the conditions that if $f(v) = 0$, then vertex $v$ must have at least two neighbors in $V_2$ or one neighbor in $V_3$, if $f(v) = 1$, then vertex $v$ must have at least one neighbor in $V_2 \cup V_3$. The weight of a double Roman dominating function $f$ is the sum $f(V) = \sum_{v \in V} f(v)$, and the double Roman domination number $\gamma_{dR}(G)$ is the minimum weight of a double Roman dominating function on $G$. A double Italian dominating function on a graph $G$ is a function $f : V \to \{0,1,2,3\}$ satisfying the condition that for every vertex $u \in V$, if $f(u) \in \{0,1\}$, then $\sum_{v \in N[u]} f(v) \ge 3$. The double Roman domination number $\gamma_{dI}(G)$ is the minimum weight of a double Italian dominating function on $G$. Mojdeh and Volkmann [D.A. Mojdeh and L. Volkmann, Roman {3}-domination (double Italian domination), Discrete Appl. Math. 283 (2020), 555--564] proved that $\gamma_{dI}(T) = \gamma_{dR}(T)$ for any tree $T$. However, we find that there is a minor issue in the proof. In this paper, we first prove that $\gamma_{dI}(T) \neq \gamma_{dR}(T)$. Subsequently, we present a sharp bound on the double Italian domination number of any non-trivial tree $T$, and characterize the trees attaining this bound.