Skip to main content
SpringerLink
Log in

Tangle and Ultrafilter: Game Theoretical Interpretation

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

This paper extends the concept of filter on X into (X, f), where X is a finite underlying set and f is a symmetric submodular function from \(2^X\) to N. Then, we show a cryptomorphism between a free ultrafilter and co-tangle on (X, f). The paper also provides game-theoretical interpretations of a branch decomposition tree and a free ultrafilter on (X, f).

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Adler, I.: Marshals, monotone marshals, and hypertree-width. J. Graph Theory 47(4), 275–296 (2004)

    Article  MathSciNet  Google Scholar 

  2. Clark, B., Whittle, G.: Tangles, trees, and flowers. J. Comb. Theory Ser. B 103(3), 385–407 (2013)

    Article  MathSciNet  Google Scholar 

  3. Diestel, R.: Ends and tangles (2015). arXiv preprint arXiv:1510.04050v2

  4. Diestel, R., Oum, S.I.: Unifying duality theorems for width parameters in graphs and matroids. I. Weak and strong duality (2014). arXiv preprint arXiv:1406.3797

  5. Geelen, J., Gerards, B., Robertson, N., Whittle, G.: Obstructions to branch-decomposition of matroids. J. Comb. Theory Ser. B 96(4), 560–570 (2006)

    Article  MathSciNet  Google Scholar 

  6. Grohe, M.: Tangled up in blue (a survey on connectivity, decompositions, and tangles) (2016). arXiv preprint arXiv:1605.06704

  7. Reed, B.: Tree width and tangles: a new connectivity measure and some applications. Surv. Comb. 241, 87–162 (1997)

    MathSciNet  MATH  Google Scholar 

  8. Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory Ser. B 52(2), 153–190 (1991)

    Article  MathSciNet  Google Scholar 

  9. Seymour, P.D., Thomas, R.: Graph searching and a min-max theorem for tree-width. J. Comb. Theory Ser. B 58(1), 22–33 (1993)

    Article  MathSciNet  Google Scholar 

  10. Sikorski, R.: Boolean Algebras, 3rd edn. Springer, Berlin (1969)

    Book  Google Scholar 

  11. Yamazaki, K.: Tangle and maximal ideal. In: Algorithms and Computation, 11th International Conference and Workshops, WALCOM 2017, Lecture Notes in Computer Science, vol. 10167, pp. 81–92. Springer (2017)

Download references

Acknowledgements

This work was supported by JSPS KAKENHI Grant number 15K00007.

Author information

Authors and Affiliations

  1. Department of Electronics and Informatics, Faculty of Science and Technology, Gunma University, 1-5-1 Tenjin-cho, Kiryu, Gunma, 376-8515, Japan

    Takaaki Fujita & Koichi Yamazaki

Authors
  1. Takaaki Fujita
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Koichi Yamazaki
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Koichi Yamazaki.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fujita, T., Yamazaki, K. Tangle and Ultrafilter: Game Theoretical Interpretation. Graphs and Combinatorics 36, 319–330 (2020). https://doi.org/10.1007/s00373-019-02085-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-019-02085-4

Keywords

Search

Navigation