On the complexity of deciding whether the distinguishing chromatic number of a graph is at most two

In an article Cheng (2009) [3] published recently in this journal, it was shown that when k≥3k≥3, the problem of deciding whether the distinguishing chromatic number of a graph is at most kk is NP-hard. We consider the problem when k=2k=2. In regards to the issue of solvability in polynomial time, we show that the problem is at least as hard as graph automorphism, but no harder than graph isomorphism.