Inverse double Roman domination in graphs

Abstract

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] such that each vertex [Formula: see text] with [Formula: see text] is adjacent to at least two vertices labeled [Formula: see text] or one vertex labeled [Formula: see text] and each vertex [Formula: see text] with [Formula: see text] is adjacent to at least one vertex [Formula: see text] with [Formula: see text]. The weight of [Formula: see text] is the sum of all labelings [Formula: see text] and is denoted by [Formula: see text]. If [Formula: see text] is a DRDF on [Formula: see text] with minimum weight [Formula: see text], then its inverse double Roman dominating function (IDRDF) [Formula: see text] is a DRDF on [Formula: see text], such that [Formula: see text], where [Formula: see text]. The inverse double Roman domination number (IDRDN) of [Formula: see text], denoted by [Formula: see text] is the minimum weight of such a function. We introduce this new type of inverse dominating function, obtain some bounds for the IDRDN of [Formula: see text]. We characterize the graphs having [Formula: see text] and the highest. We also present an approach for constructing graphs with the desired IDRDN.