On computing the distinguishing and distinguishing chromatic numbers of interval graphs and other results

A vertex k -coloring of graph G is distinguishing if the only automorphism of G that preserves the colors is the identity map. It is proper-distinguishing if the coloring is both proper and distinguishing. The distinguishing number of G , D ( G ) , is the smallest integer k so that G has a distinguishing k -coloring; the distinguishing chromatic number of G , χ D ( G ) , is defined similarly. been shown recently that the distinguishing number of a planar graph can be determined efficiently by counting a related parameter–the number of inequivalent distinguishing colorings of the graph. In this paper, we demonstrate that the same technique can be used to compute the distinguishing number and the distinguishing chromatic number of an interval graph. We make use of PQ-trees, a classic data structure that has been used to recognize and test the isomorphism of interval graphs; our algorithms run in O ( n 3 log 3 n ) time for graphs with n vertices. We also prove a number of results regarding the computational complexity of determining a graph’s distinguishing chromatic number.