On the double Roman bondage numbers of graphs

Abstract

For a graph [Formula: see text], a double Roman dominating function (DRDF) is a function [Formula: see text] having the property that if [Formula: see text] for some vertex [Formula: see text], then [Formula: see text] has at least two neighbors assigned [Formula: see text] under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text], and if [Formula: see text] then [Formula: see text] has at least one neighbor [Formula: see text] with [Formula: see text]. The weight of a DRDF [Formula: see text] is the sum [Formula: see text]. The minimum weight of a DRDF on a graph [Formula: see text] is the double Roman domination number of [Formula: see text] and is denoted by [Formula: see text]. The double Roman bondage number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality among all edge subsets [Formula: see text] such that [Formula: see text]. In this paper, we study the double Roman bondage number in graphs. We determine the double Roman bondage number in several families of graphs, and present several bounds for the double Roman bondage number. We also study the complexity issue of the double Roman bondage number and prove that the decision problem for the double Roman bondage number is NP-hard even when restricted to bipartite graphs.