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Journal of Applied Mathematics
Volume 2014, Article ID 934637, 15 pages
http://dx.doi.org/10.1155/2014/934637
Research Article

Novel Techniques to Speed Up the Computation of the Automorphism Group of a Graph

1DIATEL-UPM, 28031 Madrid, Spain
2IMDEA Networks Institute, 28918 Madrid, Spain

Received 3 April 2014; Accepted 24 June 2014; Published 24 July 2014

Academic Editor: Renat Zhdanov

Copyright © 2014 José Luis López-Presa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Graph automorphism (GA) is a classical problem, in which the objective is to compute the automorphism group of an input graph. Most GA algorithms explore a search tree using the individualization-refinement procedure. Four novel techniques are proposed which increase the performance of any algorithm of this type by reducing the depth of the search tree and by effectively pruning it. We formally prove that a GA algorithm that uses these techniques correctly computes the automorphism group of an input graph. Then, we describe how these techniques have been incorporated into the GA algorithm conauto, as conauto-2.03, with at most an additive polynomial increase in its asymptotic time complexity. Using a benchmark of different graph families, we have evaluated the impact of these techniques on the size of the search tree, observing a significant reduction both when they are applied individually and when all of them are applied together. This is also reflected in a reduction of the running time, which is substantial for some graph families. Finally, we have compared the search tree size of conauto-2.03 against those of other popular GA algorithms, observing that, in most cases, conauto explores less nodes than these algorithms.