Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

  • Jannis BulianEmail author
  • Anuj DawarEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8894)


A commonly studied means of parameterizing graph problems is the deletion distance from triviality [10], which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canonisation, and thus graph isomorphism, is \(\mathsf {FPT}\) when parameterized by elimination distance to bounded degree, generalising results of Bouland et al. [2] on isomorphism parameterized by tree-depth.


Evacuation Distance Graph Isomorphism Distal Deletion Canonical Graph Bouland 
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  1. 1.
    Babai, L., Luks, E.M.: Canonical labeling of graphs. In: Proceedings of 15th ACM Symposium. Theory of Computing, pp. 171–183. ACM, New York (1983)Google Scholar
  2. 2.
    Bouland, A., Dawar, A., Kopczyński, E.: On tractable parameterizations of graph isomorphism. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 218–230. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  3. 3.
    Diestel, R.: Graph Theory. Springer, New York (2000)Google Scholar
  4. 4.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (2012)Google Scholar
  5. 5.
    Evdokimov, S., Ponomarenko, I.: Isomorphism of coloured graphs with slowly increasing multiplicity of Jordan blocks. Combinatorica 19(3), 321–333 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Proceedings 19th International Symposium Algorithms and Computation, pp. 294–305 (2008)Google Scholar
  7. 7.
    Filotti, I.S., Mayer, J.N.: A polynomial-time algorithm for determining the isomorphism of graphs of fixed genus. In: STOC ’80: Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, ACM Request Permissions, April 1980Google Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)Google Scholar
  9. 9.
    Grohe, M., Marx, D.: Structure theorem and isomorphism test for graphs with excluded topological subgraphs. In: Proceedings 44th Symposium on Theory of Computing, pp. 173–192 (2012)Google Scholar
  10. 10.
    Guo, J., Hüffner, F., Niedermeier, R.: A structural view on parameterizing problems: distance from triviality. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 162–173. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Kratsch, S., Schweitzer, P.: Isomorphism for graphs of bounded feedback vertex set number. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 81–92. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    Lindell, S.: A logspace algorithm for tree canonization (extended abstract). In: STOC ’92: Proceedings of the Twenty-fourth Annual ACM Symposium on Theory of Computing, ACM Request Permissions, July 1992Google Scholar
  13. 13.
    Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth. arxiv:1404.0818 [cs.DS] (2014)
  14. 14.
    Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25(1), 42–65 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    McKay, B.D.: Practical graph isomorphism. Congressus Numerantium 30, 45–87 (1981)MathSciNetGoogle Scholar
  16. 16.
    McKay, B.D., Piperno, A.: Practical graph isomorphism. II. J. Symb. Comput. 60, 94–112 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Miller, G.: Isomorphism testing for graphs of bounded genus. In: STOC ’80: Proceedings 12th ACM Symposium Theory of Computing. ACM (1980)Google Scholar
  18. 18.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  19. 19.
    Ponomarenko, I.N.: The isomorphism problem for classes of graphs closed under contraction. J. Sov. Math. 55(2), 1621–1643 (1991)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Toda, S.: Computing automorphism groups of chordal graphs whose simplicial components are of small size. IEICE Trans. Inf. Syst. E89–D(8), 2388–2401 (2006)CrossRefGoogle Scholar
  21. 21.
    Uehara, R., Toda, S., Nagoya, T.: Graph isomorphism completeness for chordal bipartite graphs and strongly chordal graphs. Discrete Appl. Math. 145(3), 479–482 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Yamazaki, K., Bodlaender, H.L., De Fluiter, B., Thilikos, D.M.: Isomorphism for. In: Bongiovanni, G., Bovet, D.P., Di Battista, G. (eds.) CIAC 1997. LNCS, vol. 1203, pp. 276–287. Springer, Heidelberg (1997)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of Cambridge Computer LaboratoryCambridgeUK

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