The distinguishing chromatic number χD(G) of a graph G is the smallest number of colors needed to properly color the vertices of G such that the only automorphism of G that preserves colors is the identity. Studying the distinguishing chromatic number of graphs produced some interesting work, and in continuation, we may prefer to investigate the distinguishing chromatic number of the hierarchical product of graphs. The paper addresses the question of Choi, Hartke, and Kaule as to whether there are graphs for which the distinguishing chromatic number is near the chromatic number.

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