On Maximum Symmetric Subgraphs

  • Ho-Lin Chen
  • Hsueh-I. Lu
  • Hsu-Chun Yen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1984)


Let G be an n-node graph. We address the problem of computing a maximum symmetric graph H from G by deleting nodes, deleting edges, and contracting edges. This NP-complete problem arises naturally from the objective of drawing G as symmetrically as possible. We show that its tractability for the special cases of G being a plane graph, an ordered tree, and an unordered tree, depends on the type of operations used to obtain H from G. Moreover, we give an O(log n)-approximation algorithm for the intractable case that H is obtained from a tree G by contracting edges. As a by-product, we give an O(log n)-approximation algorithm for an NP-complete edit-distance problem.


  1. 1.
    T. Akutsu, An RNC algorithm for finding a largest common subtree of two trees, IEICE Transactions on Information Systems, E75-D, pp. 95–101, 1992.Google Scholar
  2. 2.
    S. Bachl, Isomorphic Subgraphs, International Symposium on Graph Drawing (GD’99), LNCS 1731, pp. 286–296, 1999.Google Scholar
  3. 3.
    G. Di Battista, P. Eades, and R. Tamassia and I. Tollis, Graph Drawing: Algorithms for the Visualization of Graphs, Prentice-Hall, 1999.Google Scholar
  4. 4.
    K. Chin and H. Yen, The Symmetry Number Problem for Trees, manuscript, 1998.Google Scholar
  5. 5.
    F. de Fraysseix, A Heuristic for Graph Symmetry Detection, International Symposium on Graph Drawing (GD’99), LNCS 1731, pp. 276–285, 1999.Google Scholar
  6. 6.
    P. Eades and X. Lin, Spring Algorithms and Symmetry, Computing and Combinatorics (COCOON’97), LNCS 1276, pp. 202–211, 1997.Google Scholar
  7. 7.
    M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York, 1979.MATHGoogle Scholar
  8. 8.
    S. Hong, P. Eades, and S. Lee, Finding Planar Geometric Automorphisms in Planar Graphs, 9th International Symposium on Algorithms and Computation (ISAAC’98) LNCS 1533, Springer, pp. 277–286, 1998.Google Scholar
  9. 9.
    S. Hong, P. Eades, A. Quigley, and S. Lee, Drawing Algorithms for Series Parallel Digraphs in Three Dimensions, International Symposium on Graph Drawing (GD’98), LNCS 1547, pp. 198–209, 1998.Google Scholar
  10. 10.
    S. Hong, P. Eades, A. Quigley, and S. Lee, Drawing Series-Parallel Digraphs Symmetrically, manuscript, 1999.Google Scholar
  11. 11.
    J. Hopcroft and R. Tarjan, A V 2 Algorithm for Determining Isomorphism of Planar Graphs, Information Processing Letters, 1(1):32–34, 1971.MATHCrossRefGoogle Scholar
  12. 12.
    P. Kilpelainen, and H. Mannila, Ordered and Unordered Tree Inclusion, SIAM Journal on Computing 24(2):340–356, 1995.CrossRefMathSciNetGoogle Scholar
  13. 13.
    P. Klein, Computing the Edit Distance Between Unrooted Ordered Trees, 6th European Symposium on Algorithms (ESA’98), LNCS 1461, 91–102, 1998.Google Scholar
  14. 14.
    E. Lawer, Combinatorial Optimization: Networks and Matroids, New York: Holt, Rinehart & Winston, 1976.Google Scholar
  15. 15.
    J. Manning, Geometric Symmetry in Graphs, Ph.D. Dissertation, Department of Computer Science, Purdue University, 1990.Google Scholar
  16. 16.
    J. Manning and M. Atallah. Fast Detection and Display of Symmetry in Trees, Congressus Numerantium, 64:159–169, 1988.MathSciNetGoogle Scholar
  17. 17.
    J. Manning, M. Atallah, K. Cudjoe, J. Lozito, and R. Pacheco. A System for Drawing Graphs with Geometric Symmetry, International Symposium on Graph Drawing (GD’95), LNCS 894, pp. 262–265, 1995.Google Scholar
  18. 18.
    C. Papadimitriou, Computational Complexity, Addison-Wesley, 1994.Google Scholar
  19. 19.
    K. Zhang, and T. Jiang, Some MAX SNP-hard Results Concerning Unordered Labeled Trees, Information Processing Letters 49(5):249–254, 1994.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    K. Zhang, and D. Shasha, Simple Fast Algorithms for the Editing Distance between Trees and Related Problems, SIAM Journal on Computing 18(6):1245–1262, 1989.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Ho-Lin Chen
    • 1
  • Hsueh-I. Lu
    • 2
  • Hsu-Chun Yen
    • 1
  1. 1.Department of Electrical EngineeringNational Taiwan UniversityTaipeiR.O.C.
  2. 2.Institute of Information ScienceAcademia SinicaTaipeiR.O.C.

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