The dominated coloring of a graph G is a proper coloring of G such that each color class is dominated by at least one vertex. The dominated chromatic number of a graph G is the smallest number of colors needed to color the vertices of G by this way, denoted by dom. In this paper, we de ne the covering set related to dom as a new concept. For a minimum dominated coloring of G, a set of vertices S is called a covering set of dominated coloring if each color class is dominated by a vertex of S. We call the minimum cardinality of a covering set of dominated coloring of G, covering number and we denote by dom. We obtain some bounds on dom and finally we study about covering number of subdivision, middle and total graph of paths and cycles.
