global dominator chromatic number of certain graphs

For a graph g = (v, e) and a vertex subset d ⊆ v , a vertex v ∈ v is called a dominator of d if v is adjacent to every vertex in d, and an anti-dominator of d if v is not adjacent to any vertex in d. given a coloring c = {v1, v2, . . . , vk} of g, a color class vi is a dominating color class (resp. an anti dominating color class) for a vertex v if v dominates all vertices in vi (resp. v dominates no vertex in vi). a coloring c is a global dominator coloring if each vertex in g has both a dominating and an anti-dominating color class. the global dominator chromatic number, denoted by χgd(g), is the minimum number of colors required for a global dominator coloring of g. in this paper, we investigate the global dominator chromatic number for various classes of graphs.