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Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree

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

Abstract

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.

Keywords

Evacuation Distance Graph Isomorphism Distal Deletion Canonical Graph Bouland 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  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

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of Cambridge Computer LaboratoryCambridgeUK

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