Restrained k-rainbow reinforcement number in graphs

Abstract

Let [Formula: see text] be a positive integer. A restrained [Formula: see text]-rainbow dominating function (RkRD-function) of a graph [Formula: see text] is a function [Formula: see text] from the vertex set [Formula: see text] to the set of all subsets of the set [Formula: see text] satisfying: (i) for any vertex [Formula: see text] with [Formula: see text] the condition [Formula: see text] is fulfilled, where [Formula: see text] is the set of all vertices adjacent to [Formula: see text] (ii) the subgraph induced by the vertices assigned [Formula: see text] under [Formula: see text] has no isolated vertices. The weight of an RkRD-function [Formula: see text] is the value [Formula: see text], and the restrained [Formula: see text]-rainbow domination number of [Formula: see text], denoted by [Formula: see text], is the minimum weight of an RkRD-function of [Formula: see text]. In this paper, we continue the study of the restrained [Formula: see text]-rainbow reinforcement number [Formula: see text] of a graph [Formula: see text] defined as the cardinality of a smallest set of edges that we must add to [Formula: see text] to decrease [Formula: see text] We shall first show that the decision problem associated with [Formula: see text] is NP-hard for arbitrary graphs. Then several properties as well as some sharp bounds of the restrained [Formula: see text]-rainbow reinforcement number are investigated.