Equitable K-Fair Domination under Some Binary Operations

A subset S of the vertex set V(G) of a graph G is called an equitable fair dominating set of G if S is an equitable dominating set of G and for any v, w is an element of V(G) \ S, NG(v) f1 S = NG(w) f1S >= 1. The equitable fair domination number of G, denoted by gamma efd(G), is the minimum cardinality of an EFD-set of G. The set S is called an equitable k-fair dominating set (abbreviated EkFD-set) of G if NG(v) f1S = k for any v is an element of V(G)\S, where k is a positive integer. The equitable k-fair domination number of G, denoted by gamma kefd(G), is the minimum cardinality of an EkFD-set. An equitable k-fair dominating set of cardinality gamma kefd(G) is called a gamma kefd-set of G. In this paper, we characterize the notions of equitable k-fair domination in graphs, study the EkFD-sets under some binary operations of graphs, and determine exact values or bounds for this domination variant.