  This paper serves as the first extension of the topic of dominator colorings
of graphs to the setting of digraphs. We establish the dominator chromatic
number over all possible orientations of paths and cycles. In this endeavor we
discover that there are infinitely many counterexamples of a graph and subgraph
pair for which the subgraph has a larger dominator chromatic number than the
larger graph into which it embeds. Finally, a new graph invariant measuring the
difference between the dominator chromatic number of a graph and the chromatic
number of that graph is established and studied. The paper concludes with some
of the possible avenues for extending this line of research.
