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

Abstract

a b s t r a c tIn an article Cheng (2009) [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.

Distinguishing Trees in Linear Time

“…We have shown that the distinguishing number of trees and forests can be computed in linear time, improving the previously known O(n log n) time algorithm. We believe that our algorithmic technique in Section 2 can be applied to improve by a logarithmic factor (caused by a binary search in the last step of the algorithms) the complexities of computing distinguishing numbers and distinguishing chromatic numbers of the following graph classes: (1) the distinguishing number of (i) planar graphs computed by Arvind et al [4,5] and (ii) interval graphs computed by Cheng [11]; (2) the distinguishing chromatic number (due to Collins and Trenk [12], see also [13]) of: (i) trees computed by Cheng [11] and (ii) interval graphs computed by Cheng [11].…”

Graphs with Large Distinguishing Chromatic Number

“…Clearly, any graph G of order n 2 must satisfy χ D (G) 2. Characterizing graphs with χ D (G) = 2 is an interesting but hard open problem (see [7]). In this paper, we investigate graphs with large distinguishing chromatic number and in Section 3, we characterize the graphs G of order n that satisfy χ D (G) = n − 1 or χ D (G) = n − 2.…”