Skip to main content
SpringerLink
Log in

Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10976))

Included in the following conference series:

  • 1407 Accesses

Abstract

The Dulmage-Mendelsohn decomposition is a classical canonical decomposition in matching theory applicable for bipartite graphs and is famous not only for its application in the field of matrix computation, but also for providing a prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn decomposition is stated and proved using the two color classes of a bipartite graph, and therefore generalizing this decomposition for nonbipartite graphs has been a difficult task. In this paper, we obtain a new canonical decomposition that is a generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs using a recently introduced tool in matching theory, the basilica decomposition. Our result enables us to understand all known canonical decompositions in a unified way. Furthermore, we apply our result to derive a new theorem regarding barriers. The duality theorem for the maximum matching problem is the celebrated Berge formula, in which dual optimizers are known as barriers. Several results regarding maximal barriers have been derived by known canonical decompositions; however, no characterization has been known for general graphs. In this paper, we provide a characterization of the family of maximal barriers in general graphs, in which the known results are developed and unified.

N. Kita—Supported partly by JSPS KAKENHI Grant Number 15J09683.

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

References

  1. Duff, I.S., Erisman, A.M., Reid, J.K.: Direct Methods for Sparse Matrices. Clarendon Press, Oxford (1986)

    MATH  Google Scholar 

  2. Dulmage, A.L., Mendelsohn, N.S.: Coverings of bipartite graphs. Can. J. Math. 10(4), 516–534 (1958)

    MathSciNet  MATH  Google Scholar 

  3. Dulmage, A.L., Mendelsohn, N.S.: A structure theory of bi-partite graphs. Trans. Roy. Soc. Can. Sect. 3 53, 1–13 (1959)

    MATH  Google Scholar 

  4. Dulmage, A.L., Mendelsohn, N.S.: Two algorithms for bipartite graphs. J. Soc. Ind. Appl. Math. 11(1), 183–194 (1963)

    Article  MathSciNet  Google Scholar 

  5. Edmonds, J.: Paths, trees and flowers. Can. J. Math. 17, 449–467 (1965)

    Article  MathSciNet  Google Scholar 

  6. Gallai, T.: Maximale systeme unabhängiger kanten. A Magyer Tudományos Akadémia: Intézetének Közleményei 8, 401–413 (1964)

    MathSciNet  MATH  Google Scholar 

  7. Iwata, S.: Block triangularization of skew-symmetric matrices. Linear Algebra Appl. 273(1–3), 215–226 (1998)

    Article  MathSciNet  Google Scholar 

  8. Kita, N.: New canonical decomposition in matching theory (under review). arXiv preprint arXiv:1708.01051

  9. Kita, N.: A partially ordered structure and a generalization of the canonical partition for general graphs with perfect matchings. In: Chao, K.-M., Hsu, T., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 85–94. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-35261-4_12

    Chapter  MATH  Google Scholar 

  10. Kita, N.: A partially ordered structure and a generalization of the canonical partition for general graphs with perfect matchings. CoRR abs/1205.3 (2012)

    Chapter  Google Scholar 

  11. Kita, N.: Disclosing barriers: a generalization of the canonical partition based on Lovász’s formulation. In: Widmayer, P., Xu, Y., Zhu, B. (eds.) COCOA 2013. LNCS, vol. 8287, pp. 402–413. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03780-6_35

    Chapter  Google Scholar 

  12. Kita, N.: A graph theoretic proof of the tight cut lemma (2015). arXiv preprint arXiv:1512.08870. In: The Proceedings of the 11th Annual International Conference on Combinatorial Optimization and Applications (COCOA 2017) (2017). https://doi.org/10.1007/978-3-319-71150-8_20

    Chapter  Google Scholar 

  13. Kotzig, A.: Z teórie konečných grafov s lineárnym faktorom. I (in Slovak). Math. Slovaca 9(2), 73–91 (1959)

    Google Scholar 

  14. Kotzig, A.: Z teórie konečných grafov s lineárnym faktorom. II (in Slovak). Math. Slovaca 9(3), 136–159 (1959)

    MathSciNet  Google Scholar 

  15. Kotzig, A.: Z teórie konečných grafov s lineárnym faktorom. III (in Slovak). Math. Slovaca 10(4), 205–215 (1960)

    Google Scholar 

  16. Lovász, L.: On the structure of factorizable graphs. Acta Math. Hung. 23(1–2), 179–195 (1972)

    Article  MathSciNet  Google Scholar 

  17. Lovász, L., Plummer, M.D.: Matching Theory, vol. 367. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  18. Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, New York (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Tokyo University of Science, 2641 Yamazaki, Noda, Chiba, 278-8510, Japan

    Nanao Kita

Authors
  1. Nanao Kita
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nanao Kita .

Editor information

Editors and Affiliations

  1. City University of Hong Kong, Hong Kong, China

    Lusheng Wang

  2. Shandong University, Jinan, China

    Daming Zhu

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer International Publishing AG, part of Springer Nature

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kita, N. (2018). Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality. In: Wang, L., Zhu, D. (eds) Computing and Combinatorics. COCOON 2018. Lecture Notes in Computer Science(), vol 10976. Springer, Cham. https://doi.org/10.1007/978-3-319-94776-1_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-94776-1_25

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-94775-4

  • Online ISBN: 978-3-319-94776-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation