Amortized \(\tilde{O}(|V|)\)-Delay Algorithm for Listing Chordless Cycles in Undirected Graphs

  • Rui Ferreira
  • Roberto Grossi
  • Romeo Rizzi
  • Gustavo Sacomoto
  • Marie-France Sagot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

Chordless cycles are very natural structures in undirected graphs, with an important history and distinguished role in graph theory. Motivated also by previous work on the classical problem of listing cycles, we study how to list chordless cycles. The best known solution to list all the C chordless cycles contained in an undirected graph G = (V,E) takes O(|E|2 + |E| ·C) time. In this paper we provide an algorithm taking \(\tilde{O}(|E| + |V| \cdot C)\) time. We also show how to obtain the same complexity for listing all the P chordless st-paths in G (where C is replaced by P).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Birmelé, E., Ferreira, R.A., Grossi, R., Marino, A., Pisanti, N., Rizzi, R., Sacomoto, G.: Optimal listing of cycles and st-paths in undirected graphs. In: SODA 2013, pp. 1884–1896. ACM/SIAM (2013)Google Scholar
  2. 2.
    Chen, Y., Flum, J.: On parameterized path and chordless path problems. In: IEEE Conference on Computational Complexity, pp. 250–263 (2007)Google Scholar
  3. 3.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Annals of Mathematics 164, 51–229 (2006)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Conforti, M., Cornuéjols, G., Kapoor, A., Vuskovic, K.: Recognizing balanced 0, +/- matrices. In: SODA 1994, pp. 103–111. ACM/SIAM (1994)Google Scholar
  5. 5.
    Conforti, M., Cornuéjols, G., Kapoor, A., Vuskovic, K.: Finding an even hole in a graph. In: FOCS 1997, pp. 480–485. IEEE Computer Society (1997)Google Scholar
  6. 6.
    Conforti, M., Rao, M.R.: Structural properties and decomposition of linear balanced matrices. Math. Program. 55, 129–168 (1992)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Haas, R., Hoffmann, M.: Chordless paths through three vertices. Theoretical Computer Science 351(3), 360–371 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Kapron, B.M., King, V., Mountjoy, B.: Dynamic graph connectivity in polylogarithmic worst case time. In: SODA, pp. 1131–1142 (2013)Google Scholar
  9. 9.
    Kawarabayashi, K.-I., Kobayashi, Y.: The induced disjoint paths problem. In: Lodi, A., Panconesi, A., Rinaldi, G. (eds.) IPCO 2008. LNCS, vol. 5035, pp. 47–61. Springer, Heidelberg (2008)Google Scholar
  10. 10.
    Read, C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5(3), 237–252 (1975)MATHMathSciNetGoogle Scholar
  11. 11.
    Seinsche, D.: On a property of the class of n-colorable graphs. Journal of Combinatorial Theory, Series B 16(2), 191–193 (1974)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Sokhn, N., Baltensperger, R., Bersier, L.-F., Hennebert, J., Ultes-Nitsche, U.: Identification of chordless cycles in ecological networks. In: Glass, K., Colbaugh, R., Ormerod, P., Tsao, J. (eds.) Complex 2012. LNICST, vol. 126, pp. 316–324. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  13. 13.
    Maciej, M.: Syslo. An efficient cycle vector space algorithm for listing all cycles of a planar graph. SIAM J. Comput. 10(4), 797–808 (1981)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Uno, T.: Algorithms for enumerating all perfect, maximum and maximal matchings in bipartite graphs. In: Leong, H.-V., Jain, S., Imai, H. (eds.) ISAAC 1997. LNCS, vol. 1350, pp. 92–101. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  15. 15.
    Uno, T.: An output linear time algorithm for enumerating chordless cycles. In: 92nd SIGAL of Information Processing Society Japan, pp. 47–53 (2003) (in Japanese)Google Scholar
  16. 16.
    Wild, M.: Generating all cycles, chordless cycles, and hamiltonian cycles with the principle of exclusion. J. of Discrete Algorithms 6(1), 93–102 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Rui Ferreira
    • 1
  • Roberto Grossi
    • 2
  • Romeo Rizzi
    • 3
  • Gustavo Sacomoto
    • 4
    • 5
  • Marie-France Sagot
    • 4
    • 5
  1. 1.Microsoft BingUK
  2. 2.Università di PisaItaly
  3. 3.Università di VeronaItaly
  4. 4.INRIA Grenoble Rhône-AlpesFrance
  5. 5.UMR CNRS 5558 - LBBE, Université Lyon 1France

Personalised recommendations