Restrained Italian bondage number in graphs

Abstract

A restrained Italian dominating function (RIDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying: (i) [Formula: see text] for every vertex [Formula: see text] with [Formula: see text], where [Formula: see text] is the set of vertices adjacent to [Formula: see text]; (ii) the subgraph induced by the vertices assigned 0 under [Formula: see text] has no isolated vertices. The weight of an RIDF is the sum of its function values over the whole set of vertices, and the restrained Italian domination number [Formula: see text] is the minimum weight of an RIDF on [Formula: see text] In this paper, we initiate the study of the restrained Italian bondage number [Formula: see text] of a graph [Formula: see text] with no isolated vertices defined as the smallest size of set of edges [Formula: see text] for which [Formula: see text]. We begin by showing that the decision problem associated with the restrained Italian bondage problem is NP-hard. Then basic properties of the restrained Italian bondage number are presented. Finally, some sharp bounds for [Formula: see text] are also established.