The [formula omitted]-rainbow reinforcement numbers in graphs

Abstract

Let k≥1 be an integer, and let G be a graph. A k-rainbow dominating function (or a k-RDF) of G is a function f from the vertex set V(G) to the family of all subsets of {1,2,…,k} such that for every v∈V(G) with f(v)=0̸, the condition ⋃u∈NG(v)f(u)={1,2,…,k} is fulfilled, where NG(v) is the open neighborhood of v. The weight of a k-RDF f of G is the value ω(f)=∑v∈V(G)∣f(v)∣. The k-rainbow domination number of G, denoted by γrk(G), is the minimum weight of a k-RDF of G. The 1-rainbow domination is the same as the classical domination.The k-rainbow reinforcement number of G, denoted by rrk(G), is the minimum number of edges that must be added to G in order to decrease the k-rainbow domination number. In this paper, we study the k-rainbow reinforcement number of graphs to compare γrk and γrk′ for k≠k′, and present some sharp bounds concerning the invariant.