Maximal independent sets in minimum colorings

Every graph GG contains a minimum vertex-coloring with the property that at least one color class of the coloring is a maximal independent set (equivalently, a dominating set) in GG. Among all such minimum vertex-colorings of the vertices of GG, a coloring with the maximum number of color classes that are dominating sets in GG is called a dominating-χχ-coloring of GG. The number of color classes that are dominating sets in a dominating-χχ-coloring of GG is defined to be the dominating-χχ-color number of GG. In this paper, we continue to investigate the dominating-χχ-color number of a graph first defined and studied in [1].