Mine ’Em All: A Note on Mining All Graphs

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9575)


We study the complexity of the problem of enumerating all graphs with frequency at least 1 and computing their support. We show that there are hereditary classes of graphs for which the complexity of this problem depends on the order in which the graphs should be enumerated (e.g. from frequent to infrequent or from small to large). For instance, the problem can be solved with polynomial delay for databases of planar graphs when the enumerated graphs should be output from large to small but it cannot be solved even in incremental-polynomial time when the enumerated graphs should be output from most frequent to least frequent (unless P=NP).


  1. 1.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006)MATHGoogle Scholar
  2. 2.
    Garey, M.R., Johnson, D.S., Tarjan, R.E.: The planar hamiltonian circuit problem is np-complete. SIAM J. Comput. 5(4), 704–714 (1976)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Grohe, M., Marx, D.: Structure theorem and isomorphism test for graphs with excluded topological subgraphs. CoRR, abs/1111.1109 (2011)Google Scholar
  4. 4.
    Horváth, T., Otaki, K., Ramon, J.: Efficient frequent connected induced subgraph mining in graphs of bounded tree-width. In: Blockeel, H., Kersting, K., Nijssen, S., Železný, F. (eds.) ECML PKDD 2013, Part I. LNCS, vol. 8188, pp. 622–637. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  5. 5.
    Horváth, T., Ramon, J.: Efficient frequent connected subgraph mining in graphs of bounded tree-width. Theoret. Comput. Sci. 411(31–33), 2784–2797 (2010)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Johnson, D.S., Yannakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Inf. Process. Lett. 27(3), 119–123 (1988)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lokshtanov, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth. In: 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, pp. 186–195 (2014)Google Scholar
  8. 8.
    Luks, E.M.: Isomorphism of graphs of bounded valence can be tested in polynomial time. J. Comput. Syst. Sci. 25(1), 42–65 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Marx, D., Pilipczuk, M.: Everything you always wanted to know about the parameterized complexity of subgraph isomorphism (but were afraid to ask). In: 31st International Symposium on Theoretical Aspects of Computer Science, STACS 2014, pp. 542–553 (2014)Google Scholar
  10. 10.
    Matoušek, J., Thomas, R.: On the complexity of finding iso- and other morphisms for partial k-trees. Discrete Math. 108(1–3), 343–364 (1992)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Robertson, N., Seymour, P.D.: Graph minors. XIII. the disjoint paths problem. J. Comb. Theory Ser. B 63(1), 65–110 (1995)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Torán, J., Wagner, F.: The complexity of planar graph isomorphism. Bull. EATCS 97, 60–82 (2009)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.School of Computer Science & InformaticsCardiff UniversityCardiffUK
  2. 2.Department of Computer ScienceKU LeuvenLeuvenBelgium
  3. 3.INRIALilleFrance

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