Computing and Drawing Isomorphic Subgraphs

  • Sabine Bachl
  • Franz-Josef Brandenburg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2528)


The isomorphic subgraph problem is finding two disjoint subgraphs of a graph which coincide on at least k edges. Then the graph partitions into a large subgraph, its copy and a remainder. The problem resembles the NP-hard largest common subgraph problem. In [1],[2] it has been shown that the isomorphic subgraph problem is NP-hard, even for restricted instances. In this paper we present a greedy heuristic for the approximation of large isomorphic subgraphs and introduce a spring algorithm which preserves isomorphic subgraphs and displays them as copies of each other. The heuristic has been tested extensively on four independent test suites. The drawing algorithm yields nice drawings which cannot be obtained by standard spring algorithms.


  1. 1.
    S. Bachl. Isomorphic subgraphs. Proc. Graph Drawing’99, LNCS 1731 (1999), 286–296.Google Scholar
  2. 2.
    S. Bachl. Erkennung isomorpher Subgraphen und deren Anwendung beim Zeichnen von Graphen. Dissertation, University of Passau, (2001),
  3. 3.
    L. Babai and L. Kucera. Canonical labelling of graphs in average linear time. Proc. 20th IEEE FOCS (1979), 39–46.Google Scholar
  4. 4.
    G. Di Battista, P. Eades, R. Tamassia, and I.G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice Hall, (1999).Google Scholar
  5. 5.
    T. Biedl, J. Marks, K. Ryall, and S. Whitesides. Graph multidrawing: Finding nice drawings without defining nice. Proc. Graph Drawing’98, LNCS 1547 (1998), 347–355.Google Scholar
  6. 6.
    F.J. Brandenburg. Pattern matching problems in graphs. Unpublished manuscript, (2000).Google Scholar
  7. 7.
    H.-L. Chen, H.-I. Lu and H.-C. Yen. On maximum symmetric subgraphs. Proc. Graph Drawing’00, LNCS 1984 (2001), 372–383.Google Scholar
  8. 8.
    J. Clark and D. Holton. Graphentheorie-Grundlagen und Anwendungen. Spektrum Akademischer Verlag, (1991).Google Scholar
  9. 9.
    D. Corneil and D. Kirkpatrick. A theoretical analysis of various heuristics for the graph isomorphism problem. SIAM J. Comput. 9, (1980), 281–297.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    P. Eades. Drawing free trees. Bulletin of the Institute for Combinatorics and its Applications 5, (1992), 10–36.MATHMathSciNetGoogle Scholar
  11. 11.
    P. Eades. A heuristic for graph drawing. Cong. Numer. 42, (1984), 149–160.MathSciNetGoogle Scholar
  12. 12.
    P. Eades and X. Lin. Spring algorithms and symmetry. Theoret. Comput. Sci. 240 (2000), 379–405.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Forster. Zeichnen ungerichteter Graphen mit gegebenen Knotengröβen durchein Springembedder-Verfahren. Diplomarbeit, Universität Passau, (1999).Google Scholar
  14. 14.
    A. Frick, A. Ludwig and H. Mehldau. A fast adaptive layout algorithm for undirected graphs. Proc. Graph Drawing’94, LNCS 894 (1995), 388–403.Google Scholar
  15. 15.
    T. Fruchterman and E.M. Reingold. Graph drawing by force-directed placement. Software-Practice and Experience 21, (1991), 1129–1164.CrossRefGoogle Scholar
  16. 16.
    M.R. Garey and D.S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, San Francisco, (1979).MATHGoogle Scholar
  17. 17.
    A. Gupta and N. Nishimura. The complexity of subgraph isomorphism for classes of partial k-trees. Theoret. Comput. Sci. 164, (1996), 287–298.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    S.-H. Hong, B. McKay and P. Eades. Symmetric drawings of triconnected planargraphs. Proc. 13 ACM-SIAM Symposium on Discrete Algorithms (2002), 356–365.Google Scholar
  19. 19.
    I. Koch. Enumerating all connected maximal common subgraphs in two graphs. Theoret. Comput. Sci. 250, (2001), 1–30.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    G. Levi. A note on the derivation of maximal common subgraphs of two directed or undirected graphs. Calcolo 9, (1972), 341–352.CrossRefMathSciNetGoogle Scholar
  21. 21.
    A. Lubiw. Some NP-complete problems similar to graph isomorphism. SIAM J. Comput. 10, (1981), 11–21.MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    J. Manning. Geometric symmetry in graphs. Ph.D. thesis, Purdue Univ., (1990).Google Scholar
  23. 23.
    J. Manning. Computational complexity of geometric symmetry detection in graphs. LNCS 507, (1990), 1–7.Google Scholar
  24. 25.
    H. Purchase, R. Cohen and M. James. Validating graph drawing aesthetics. Proc. Graph Drawing’95, LNCS 1027 (1996), 435–446.Google Scholar
  25. 26.
    H. Purchase. Which aesthetic has the greatest effect on human understanding. Proc. Graph Drawing’97, LNCS 1353 (1997), 248–261.Google Scholar
  26. 27.
    R.C. Read and R.J. Wilson An Atlas of Graphs. Clarendon Press Oxford (1998)MATHGoogle Scholar
  27. 28.
  28. 29.
    E.M. Reingold and J.S. Tilford. Tidier drawings of trees. IEEE Trans. SE 7, (1981), 223–228.Google Scholar
  29. 30.
    K.J. Supowit and E.M. Reingold. The complexity of drawing trees nicely. Acta Informatica 18, (1983), 377–392.MATHCrossRefMathSciNetGoogle Scholar
  30. 31.
    J.R. Ullmann. An algorithm for subgraph isomorphism. J. Assoc. Comput. Mach. 16, (1970), 31–42.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Sabine Bachl
    • 1
  • Franz-Josef Brandenburg
    • 2
  1. AGMünchenGermany
  2. 2.University of PassauPassauGermany

Personalised recommendations