Double Italian domination in trees
Weiping Shang, Shanshan Zhang

TL;DR
This paper investigates the double Italian domination number in trees, correcting a previous proof, and establishes a sharp bound with characterizations for non-trivial trees.
Contribution
It disproves a prior claim that the double Italian and double Roman domination numbers are equal in trees and provides a new bound with characterizations.
Findings
Disproved the equality of double Italian and double Roman domination numbers in trees.
Established a sharp bound for the double Italian domination number in non-trivial trees.
Characterized trees that attain this bound.
Abstract
Let be a graph with vertex set . A double Roman dominating function on a graph is a function satisfying the conditions that if , then vertex must have at least two neighbors in or one neighbor in , if , then vertex must have at least one neighbor in . The weight of a double Roman dominating function is the sum , and the double Roman domination number is the minimum weight of a double Roman dominating function on . A double Italian dominating function on a graph is a function satisfying the condition that for every vertex , if , then . The double Roman domination number is the minimum weight of a double Italian dominating function on . Mojdeh…
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