Automorphism Group of Marked Interval Graphs in FPT

Abstract

An interval graph is the intersection graph of a set of intervals on the real line. By a classical result of [Colbourn and Booth], the automorphism group of an interval graph can be computed in linear time using so-called PQ-trees. We consider the following generalization; some cliques of our interval graph are marked with colors, and we restrict to only such automorphisms that map every marked clique to a marked clique of the same color. We call such automorphisms marking preserving, and we are in fact interested in the action of their group on the marked sets. While this problem is at least as hard as general graph isomorphism (i.e., GI-hard), we give an FPT-time algorithm with respect to the parameter equal to the largest number of marked sets incomparable by inclusion. This result is useful, e.g., for the isomorphism problem of chordal graphs of bounded leafage.