dominated coloring of certain graphs

A proper coloring of a graph g is called a dominated coloring whenever each color class is dominated by at least one vertex. the minimum number of colors among all dominated colorings of g is called its dominated chromatic number, denoted by χdom(g). we define a parameter related to dominated coloring, namely dominated chromatic covering. for a minimum dominated coloring of g, a set of vertices s is called a dominated chromatic covering if each color class is dominated by a vertex of s. the minimum cardinality of a dominated chromatic covering of g is called its dominated chromatic covering number, denoted by θχdom(g). it is clear that θχdom(g) ≤ χdom(g). in this paper, we obtain the dominated chromatic number and θχdom(g) when g is middle and total graph of paths and cycles.