Dominated coloring in certain networks

A proper vertex coloring of G is a dominated coloring if each color class is dominated by at least one vertex of G, that is, for each color class there exists a vertex that is adjacent to all the vertices of the class. The dominated chromatic number \(\chi _{\textrm{dom}}^{}(G)\) ${\chi }_{\text{dom}}^{}(G)$ is the minimum number of colors needed for a dominated coloring of G. In this paper, we obtain the dominated chromatic number for circulant network, Sierpiński network, chain silicate network, and cyclic silicate network.