On some variants of connected double Italian domination

A connected double Italian dominating function is a double Italian dominating function [Formula: see text] for which the set of vertices assigned [Formula: see text] under [Formula: see text] is connected. The connected double Italian domination number denoted by [Formula: see text], represents the minimum weight among all connected double Italian dominating functions defined on [Formula: see text]. In this study, we make significant contributions to the exploration of connected double Italian domination in graphs. We demonstrate that determining the connected double Italian domination number is an NP-complete problem. Furthermore, we classify the families of all connected graphs [Formula: see text] for which [Formula: see text]. Additionally, we examine upper and lower bounds for trees and provide a detailed characterization of the Cartesian and Lexicographic products of specific graphs with respect to their connected double Italian domination number.