THE GENERALIZATION OF TOTAL k-RAINBOW DOMINATION NUMBERS IN GRAPHS

Let k ≥ ۱ be an integer, and let G be a graph with no isolated vertex. A generalized total k-rainbow dominating function (or a gtk - RDF) of G is a function f from the vertex set V(G) to the family of all subsets of {۱, ۲, ۳, .., k} such that for every v E V(G), the condition UuεNG(v) f(u) = {۱,۲,...,k} is fulfilled, where NG(v) is the open neighborhood of v. The weight of a gtk – RDF f of G is the value w(f) = ΣvEV(G)|f(v)|. The generalized total k-rainbow domination number of G, denoted by Ygtk (G), is the minimum weight of a generalized total k-rainbow dominating function on G. The generalized total k-rainbow domination is a generalization of the total k-rainbow domination that was introduced by H. A. Ahangar and et al in ۲۰۱۷. In this paper we initiate the study of generalized total k-rainbow domination number and we investigate its basic properties. In particular, we present some bounds on the generalized total k-rainbow domination number and we determine the generalized total k-rainbow domination number of some classes of graphs.

کلیدواژه ها:

domination number ، Dominating set ، Petersen graph ، Rainbow domination