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

TL;DR

This paper investigates the computational complexity of determining whether a graph's distinguishing chromatic number is at most two, revealing it is as hard as graph automorphism but not harder than graph isomorphism.

Contribution

It establishes the complexity bounds for the k=2 case, filling a gap in understanding the problem's difficulty for small k.

Findings

Deciding if the distinguishing chromatic number ≤ 2 is as hard as graph automorphism.

The problem is no harder than graph isomorphism.

For k ≥ 3, the problem is NP-hard, but for k=2, it has a complexity bound between automorphism and isomorphism.

Abstract

In an article [3] published recently in this journal, it was shown that when k >= 3, the problem of deciding whether the distinguishing chromatic number of a graph is at most k is NP-hard. We consider the problem when k = 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.