Mixed k-rainbow domination numbers in graphs

Abstract

Let k ≥ 1 be an integer. A mixed k-rainbow dominating function (MfcRDF) of a graph G = (V, E) is a function f from the set Z = V∪E to the set of all subsets of the set {1,2.....k}, i.e. f: V∪E → P({l,2,...,k}), such that for any element z ∈ Z with f(z) = θ the condition Uα∈Nm(x)f(a) = {1.2.....k} is fulfilled, where Nm(z) is the set of all elements either adjacent or incident to the element z. The weight of an MfcRDF/is the value u(f) = Σx∈Z |f(z)|. The mixed k-rainbow domination number of a graph G, denoted by (G), is the minimum weight of an MfcRDF of G. In this paper, we initiate the study of the mixed fc-rainbow domination number in graphs, and we obtain several bounds for (G). In addition, we determine the mixed fc-rainbow domination number of some classes of graphs. © 2018 Utilitas Mathematica Publishing Inc. All rights reserved.