Advertisement

Interval Completion with the Smallest Max-Degree

(Extended Abstract)
  • Fedor V. Fomin
  • Petr A. Golovach
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1517)

Abstract

The interval degree of a graph is defined to be the smallest max-degree of any of its interval supergraphs. We find various bounds for this parameter. We prove that for any graph G the interval degree of G is at least the bandwidth of G, the pathwidth of G2 and at most twice the bandwidth of G. Also we show that if G is an AT-free claw-free graph, then the interval degree of G is equal to the clique number of G2 minus one. Finally, we show that there is a polynomial time algorithm for computing the interval degree of AT-free claw-free graphs.

Keywords

Interval Graph Chordal Graph Minimal Separator Left Graph Clique Number 
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.
    Blllionnet, A.: On interval graphs and matrice profiles. RAIRO Rech. Opér. 20, 245–256 (1986)Google Scholar
  2. 2.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Technical Report UU-CS-1996-02, Department of Computer Science, Utrecht University, Utrecht, the Netherlands (1996)Google Scholar
  3. 3.
    Bondy, J.A.: Basic graph theory: Paths and circuits. In: Graham, R.L., Grötschel, M., Lovász, L. (eds.) Handbook of Combinatorics, vol. 1, pp. 3–110. Elsevier Science B.V., Amsterdam (1995)Google Scholar
  4. 4.
    Chinn, P.Z., Chvatalova, J., Dewdney, A.K., Gibbs, N.E.: The bandwidth problem for graphs and matrices — a survey. J. Graph Theory 6, 223–254 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Ellis, J.A., Sudborough, I.H., Turner, J.: The vertex separation and search number of a graph. Information and Computation 113, 50–79 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)Google Scholar
  7. 7.
    Karpinski, M., Wirtgen, J.: On approximation hardness of the bandwidth problem. Technical Report TR-97-041, ECCC (1997)Google Scholar
  8. 8.
    Klnnersley, N.G.: The vertex separation number of a graph equals its path width. Inform. Proc. Letters 42, 345–350 (1992)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Klrousis, L.M., Papadimitriou, C.H.: Interval graphs and searching. Disc. Math. 55, 181–184 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Kloks, T.: Treewidth. Computations and Approximations. LNCS, vol. 842. Springer, Berlin (1994)zbMATHGoogle Scholar
  11. 11.
    Kloks, T., Kratsch, D., Müller, H.: Approximating the bandwidth for asteroidal triple-free graphs. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 434–447. Springer, Heidelberg (1995)Google Scholar
  12. 12.
    Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64 (1962)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Möhring, R.H.: Graph problems related to gate matrix layout and PLA folding. In: Mayr, E., Noltemeier, H., Syslo, M. (eds.) Computational Graph Theory, Comuting Suppl., pp. 17–51. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  14. 14.
    Möhring, R.H.: Triangulating graphs without asteroidal triples. Disc. Appl. Math. 64, 281–287 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Monien, B.: The bandwidth minimization problem for caterpillars with hair length 3 is NP-complete. SIAM J. Alg. Disc. Meth. 7, 505–512 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    MüIler, H.: Personal communication (1998)Google Scholar
  17. 17.
    Parra, A., Scheffler, P.: Treewidth equals bandwidth for AT-free claw-free graphs, Technical Report 436/1995, Technische Universitat Berlin, Fachbereich Ma-thematik, Berlin, Germany (1995)Google Scholar
  18. 18.
    Robertson, N., Seymour, P.D.: Graph minors. I. Excluding a forest. J. Comb. Theory Series B 35, 39–61 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Rose, D.J., Tarjan, R.E., Lueker, G.S.: Algorithmic aspects of vertex elimination on graphs. SIAM J. Comput. 5, 266–283 (1976)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Petr A. Golovach
    • 2
  1. 1.Department of Operations Research, Faculty of Mathematics and MechanicsSt.Petersburg State UniversitySt.PetersburgRussia
  2. 2.Department of Applied Mathematics, Faculty of MathematicsSyktyvkar State UniversitySyktyvkarRussia

Personalised recommendations