Skip to main content
SpringerLink
Log in

On Reachability in Graphs with Bounded Independence Number

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2387))

Included in the following conference series:

Abstract

We study the reachability problem for finite directed graphs whose independence number is bounded by some constant k. This problem is a generalisation of the reachability problem for tournaments. We show that the problem is first-order definable for all k. In contrast, the reachability problems for many other types of finite graphs, including dags and trees, are not first-order definable. Also in contrast, first-order definability does not carry over to the infinite version of the problem. We prove that the number of strongly connected components in a graph with bounded independence number can be computed using TC0-circuits, but cannot be computed using AC0-circuits. We also study the succinct version of the problem and show that it is Π P2 -complete for all k.

Work done in part while visiting the University of Rochester, New York. Supported by a TU Berlin Erwin-Stephan-Prize grant.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai. Σ 11 formulae on finite structures. Annals of Pure and Applied Logic, 24:1–48, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Cook and P. McKenzie. Problems complete for deterministic logarithmic space. J. Algorithms, 8(3):385–394, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Ehrenfeucht. An application of games to the completeness problem for formalized theories. Fundamenta Mathematicae, 49:129–141, 1961.

    MATH  MathSciNet  Google Scholar 

  4. R. Faudree, R. Gould, L. Lesniak, and T. Lindquester. Generalized degree conditions for graphs with bounded independence number. J. Graph Theory, 19(3):397–409, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  5. R. Fraïssé. Sur quelques classifications des systèmes de relations. Publ. Sci. Univ. Alger. Sér. A, 1:35–182, 1954.

    Google Scholar 

  6. M. Furst, J. Saxe, and M. Sipser. Parity, circuits, and the polynomial-time hierarchy. Math. Systems Theory, 17(1):13–27, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Galperin and A. Wigderson. Succinct representations of graphs. Inform. Control, 56(3):183–198, 1983.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Gottlob, N. Leone, and H. Veith. Succinctness as a source of complexity in logical formalisms. Annals of Pure and Applied Logic, 97(1–3):231–260, 1999.

    Article  MATH  MathSciNet  Google Scholar 

  9. F. Harary and L. Moser. The theory of round robin tournaments. Amer. Math. Monthly, 73:231–246, 1966.

    Article  MATH  MathSciNet  Google Scholar 

  10. L. Hemaspaandra and L. Torenvliet. Optimal advice. Theoretical Comput. Sci., 154(2):367–377, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  11. N. Immerman. Descriptive Complexity. Springer-Verlag, 1998.

    Google Scholar 

  12. N. Jones. Space-bounded reducibility among combinatorial problems. J. Comput. Syst. Sci, l l(l):68–85, 1975.

    Article  Google Scholar 

  13. N. Jones, Y. Lien, and W. Laaser. New problems complete for nondeterministic log space. Math. Systems Theory, 10:1–17, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Landau. On dominance relations and the structure of animal societies, III: The condition for secure structure. Bull. Mathematical Biophysics, 15(2):143–148, 1953.

    Article  Google Scholar 

  15. H. Lewis and C. Papadimitriou. Symmetric space-bounded computation. Theoretical Comput. Sci., 19(2):161–187, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Lindell. A purely logical characterization of circuit uniformity. In Proc. 7th Struc. in Complexity Theory Conf., pages 185–192. IEEE Computer Society, 1992.

    Google Scholar 

  17. N. Megiddo and U. Vishkin. On finding a minimum dominating set in a tournament. Theoretical Comput. Sci., 61(2–3):307–316, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Moon. Topics on Tournaments. Holt, Rinehart, and Winston, 1968.

    Google Scholar 

  19. C. Papadimitriou and M. Yannakakis. A note on succinct representations of graphs. Inform. Control, 71(3):181–185, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  20. K. Reid and L. Beineke. Tournaments. In Selected Topics in Graph Theory, pages 169–204. Academic Press, 1978.

    Google Scholar 

  21. P. Turán. Eine Extremalaufgabe aus der Graphentheorie (in Hungarian). Matem. és Physikai Lapok, 48:436–452, 1941.

    MATH  Google Scholar 

  22. P. Turán. On the theory of graphs. Colloquium Math., 3:19–30, 1954.

    MATH  MathSciNet  Google Scholar 

  23. K. Wagner. The complexity of problems concerning graphs with regularities. In Proc. 7th Symposium on Math. Foundations of Comp. Sci., volume 176 of Lecture Notes in Computer Science, pages 544–552. Springer-Verlag, 1984.

    Google Scholar 

  24. K. Wagner. The complexity of combinatorial problems with succinct input representation. Acta Informatica, 23(3):325–356, 1986.

    Article  MATH  MathSciNet  Google Scholar 

  25. C. Wrathall. Complete sets and the polynomial-time hierarchy. Theoretical Comput. Sci., 3(1):23–33, 1976.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Elektrotechnik und Informatik, Technische Universität Berlin, 10623, Berlin, Germany

    Arfst Nickelsen & Till Tantau

Authors
  1. Arfst Nickelsen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Till Tantau
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara, California, 93106, USA

    Oscar H. Ibarra

  2. Department of Mathematics, National University of Singapore, Singapore, Singapore, 117543

    Louxin Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Nickelsen, A., Tantau, T. (2002). On Reachability in Graphs with Bounded Independence Number. In: Ibarra, O.H., Zhang, L. (eds) Computing and Combinatorics. COCOON 2002. Lecture Notes in Computer Science, vol 2387. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45655-4_59

Download citation

  • DOI: https://doi.org/10.1007/3-540-45655-4_59

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-43996-7

  • Online ISBN: 978-3-540-45655-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation