Advertisement

An \(\mathcal {O}(n^2)\) Time Algorithm for the Minimal Permutation Completion Problem

  • Christophe CrespelleEmail author
  • Anthony Perez
  • Ioan Todinca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

We provide an \(O(n^2)\) time algorithm computing a minimal permutation completion of an arbitrary graph \(G=(V,E)\), i.e., a permutation graph \(H = (V,F)\) on the same vertex set, such that \(E \subseteq F\) and F is inclusion-minimal among all possibilities.

References

  1. 1.
    Bergeron, A., Chauve, C., de Montgolfier, F., Raffinot, M.: Computing common intervals of K permutations, with applications to modular decomposition of graphs. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 779–790. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Burzyn, P., Bonomo, F., Durán, G.: NP-completeness results for edge modification problems. Discrete Appl. Math. 154(13), 1824–1844 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Crespelle, C., Paul, C.: Fully dynamic algorithm for recognition and modular decomposition of permutation graphs. Algorithmica 58(2), 405–432 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Crespelle, C., Todinca, I.: An \(O(n^{2})\)-time algorithm for the minimal interval completion problem. Theor. Comput. Sci. 494, 75–85 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Heggernes, P.: Minimal triangulations of graphs: a survey. Discrete Math. 306(3), 297–317 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Heggernes, P., Mancini, F., Papadopoulos, C.: Minimal comparability completions of arbitrary graphs. Discrete Appl. Math. 156(5), 705–718 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Heggernes, P., Telle, J.A., Villanger, Y.: Computing minimal triangulations in time \({O}(n^{\alpha \log n}) = o(n^{2.376})\). SIAM J. Discrete Math. 19(4), 900–913 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Heggernes, P., Mancini, F.: Minimal split completions. Discrete Appl. Math. 157(12), 2659–2669 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lokshtanov, D., Mancini, F., Papadopoulos, C.: Characterizing and computing minimal cograph completions. Discrete Appl. Math. 158(7), 755–764 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mancini, F.: Graph Modification Problems Related to Graph Classes. Ph.D. thesis, University of Bergen, Norway (2008)Google Scholar
  11. 11.
    Ohtsuki, T., Mori, H., Kashiwabara, T., Fujisawa, T.: On minimal augmentation of a graph to obtain an interval graph. J. Comput. Syst. Sci. 22(1), 60–97 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ohtsuki, T.: A fast algorithm for finding an optimal ordering for vertex elimination on a graph. SIAM J. Comput. 5(1), 133–145 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rapaport, I., Suchan, K., Todinca, I.: Minimal proper interval completions. Inf. Process. Lett. 5, 195–202 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5(2), 266–283 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Yannakakis, M.: Computing the minimum fill-in is NP-complete. SIAM. J. Algebraic Discrete Methods 2(1), 77–79 (1981)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Christophe Crespelle
    • 1
    Email author
  • Anthony Perez
    • 2
  • Ioan Todinca
    • 2
  1. 1.Université Claude Bernard Lyon 1 and CNRS, DANTE/INRIA, LIP UMR CNRS 5668, ENS de Lyon, Université de Lyon and Institute of Mathematics, Vietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.University of Orléans, INSA Centre Val de Loire, LIFO EA 4022OrléansFrance

Personalised recommendations