Abstarct: A fair dominating set in a graph G (or FD-set) is a dominating set D such that all vertices not in D are dominated by the same number of vertices from D; that is, every two vertices not in D have the same number of neighbors in D. The fair domination number, fd(G), of G is the minimum cardinality of a FD-set. A fair dominating set of cardinality fd(G) is called a -fd(G) set. In chapter 1, we review theory graph concepts that we need them in the future chapters. In chapter 2, we represent fair domination concepts in graphs and we prover problems about it. In chapter 3, our purpose is to present equality between the domination number and fair domination number. Therefore we define some operations and we want to prover this theorem. In chapter 4, our purpose is to present strong equality between the domination number and fair domination number. Results in chapter 3 and 4 are presented for first time in this thesis.