Advertisement

Minimal Proper Interval Completions

  • Ivan Rapaport
  • Karol Suchan
  • Ioan Todinca
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4271)

Abstract

Given an arbitrary graph G=(V,E) and a proper interval graph H=(V,F) with E ⊆ F we say that H is a proper interval completion of G. The graph H is called a minimal proper interval completion of G if, for any sandwich graph H′=(V,F′) with E ⊆ F′ ⊂ F, H′ is not a proper interval graph. In this paper we give a \({{\mathcal{O}}(n+m)}\) time algorithm computing a minimal proper interval completion of an arbitrary graph. The output is a proper interval model of the completion.

Keywords

Maximal Clique Interval Graph Minimal Separator Arbitrary Graph Interval Model 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berry, A., Bordat, J.P.: Separability Generalizes Dirac’s Theorem. Discrete Applied Mathematics 84(1-3), 43–53 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bodlaender, H.L.: A Linear-Time Algorithm for Finding Tree-Decompositions of Small Treewidth. SIAM Journal on Computing 25(6), 1305–1317 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Cai, L.: Fixed-Parameter Tractability of Graph Modification Problems for Hereditary Properties. Information Processing Letters 58(4), 171–176 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, London (1980)zbMATHGoogle Scholar
  5. 5.
    Habib, M., Paul, C., Viennot, L.: Partition Refinement Techniques: An Interesting Algorithmic Tool Kit. International Journal of Foundations of Computer Science 10(2), 147–170 (1999)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Habib, M., McConnell, R.M., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theoretical Computer Science 234(1-2), 59–84 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Heggernes, P., Mancini, F.: Minimal Split Completions of Graphs. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 592–604. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Heggernes, P., Suchan, K., Todinca, I., Villanger, Y.: Minimal Interval Completions. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 403–414. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Heggernes, P., Telle, J.A., Villanger, Y.: Computing minimal triangulations in time O(n α log n) = o(n 2.376). In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms - SODA 2005, pp. 907–916. SIAM, Philadelphia (2005)Google Scholar
  10. 10.
    Kaplan, H., Shamir, R.: Pathwidth, Bandwidth, and Completion Problems to Proper Interval Graphs with Small Cliques. SIAM Journal on Computing 25(3), 540–561 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kaplan, H., Shamir, R., Tarjan, R.E.: Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs. SIAM Journal on Computing 28(5), 1906–1922 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kratsch, D., Spinrad, J.: Minimal fill in cO(n 2.69) time. Discrete Applied Mathematics (to appear)Google Scholar
  13. 13.
    Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 in NP-complete. SIAM Journal on Algebraic and Discrete Methods 7, 505–512 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Panda, B.S., Das, S.K.: A linear time recognition algorithm for proper interval graphs. Information Processing Letters 87(3), 153–161 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rappaport, I., Suchan, K., Todinca, I.: Minimal proper interval completions. Technical Report RR-2006-02, LIFO - University of Orléans (2006), http://www.univ-orleans.fr/SCIENCES/LIFO/prodsci/rapports/RR2006.htm.en
  16. 16.
    Rose, D., Tarjan, R.E., Lueker, G.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 146–160 (1976)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Ivan Rapaport
    • 1
  • Karol Suchan
    • 2
    • 3
  • Ioan Todinca
    • 2
  1. 1.Departamento de Ingeniería Matemática and Centro de Modelamiento MatemáticoUniversidad de ChileSantiagoChile
  2. 2.LIFO, Université d’OrléansOrléansFrance
  3. 3.Department of Discrete Mathematics, Faculty of Applied MathematicsAGH – University of Science and TechnologyCracowPoland

Personalised recommendations