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A proper vertex coloring of a graph where every node of the graph dominates all nodes of some color class is called the dominator coloring of the graph. The least number of colors used in the dominator coloring of a graph is called the dominator coloring number denoted by \(X_d(G)\). The dominator coloring number and domination number of central, middle, total and line graph of quadrilateral snake graph are derived and the relation between them are expressed in this paper.