The l-distance k-rainbow domination numbers of graphs

Abstract

Let [Formula: see text] be two positive integers. An [Formula: see text]-distance [Formula: see text]-rainbow dominating function on [Formula: see text] is a function [Formula: see text] that assigns to each vertex a set of colors chosen from the set [Formula: see text] such that for any [Formula: see text], [Formula: see text] implies [Formula: see text]. The weight of an [Formula: see text]-distance [Formula: see text]-rainbow dominating function [Formula: see text] is the value [Formula: see text] over all such functions [Formula: see text]. The [Formula: see text]-distance [Formula: see text]-rainbow domination number of a graph [Formula: see text], denoted by [Formula: see text], equals the minimum weight of a [Formula: see text]-distance [Formula: see text]-rainbow dominating function on [Formula: see text]. Note that the [Formula: see text]-distance [Formula: see text]-rainbow domination number [Formula: see text] is the usual [Formula: see text]-rainbow domination number [Formula: see text]. In this study, the [Formula: see text]-distance [Formula: see text]-rainbow domination number is defined and its properties are characterized.