Dissimilarity Between Dominator and Total Dominator Coloring of Certain Graphs

Let us consider an undirected and a finite graph \(\mathcal {G}=\{\mathcal {V(G),E(G)}\} \) with no loops or parallel edges. Here \(\mathcal {V(G)}\) and \(\mathcal {E(G)}\) represent the collection of vertices and edges respectively. A Dominator Coloring of a graph \(\mathcal {G}\) is a coloring that is proper in such a way that every vertex in \(\mathcal {G}\) is in the neighborhood of all the vertices of at least a single color class. A Total Dominator Coloring of a graph \(\mathcal {G}\) is also a proper coloring in which each vertex dominates at least a color class other than its own. The Dominator coloring has already been employed for the m-shadow graph of paths. In this article, we investigate the Dominator and Total Dominator coloring for m-Shadow graphs of Cycle, Complete graph and Wheel graph and also determine the respective dominator chromatic number and compare the same.