Profiles of Separations: in Graphs, Matroids, and Beyond

Abstract

We show that all the tangles in a finite graph or matroid can be distinguished by a single tree-decomposition that is invariant under the automorphisms of the graph or matroid. This comes as a corollary of a similar decomposition theorem for more general combina- torial structures, which has further applications.

These include a new approach to cluster analysis and image segmentation. As another illustration for the abstract theorem, we show that applying it to edge-tangles yields the Gomory-Hu theorem.

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

References

  1. [1]

    J. Carmesin, R. Diestel, M. Hamann and F. Hundertmark: k-Blocks: a connectivity invariant for graphs, SIAM J. Discrete Math. 28 (2014), 1876–1891.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    J. Carmesin, R. Diestel, M. Hamann and F. Hundertmark: Canonical tree-decompositions of finite graphs I. Existence and algorithms, J. Combin. Theory Ser. B 116 (2016), 1–24.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    J. Carmesin, R. Diestel, M. Hamann and F. Hundertmark: Canonical tree-decompositions of finite graphs II. Essential parts, J. Combin. Theory Ser. B 118 (2016), 268–283.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    J. Carmesin, R. Diestel, F. Hundertmark and M. Stein: Connectivity and tree structure in finite graphs, Combinatorica 34 (2014), 1–35.

    MathSciNet  Article  MATH  Google Scholar 

  5. [5]

    R. Diestel: Graph Theory (5th edition, 2016), Springer-Verlag, 2017. Electronic edition available at http://diestel-graph-theory.com/.

    Google Scholar 

  6. [6]

    R. Diestel: Abstract separation systems, Order 35 (2018), 157–170.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    R. Diestel: Tree sets, Order 35 (2018), 171–192.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    R. Diestel, J. Erde and P. Eberenz: Duality theorem for blocks and tangles in graphs, SIAM J. Discrete Math. 31 (2017), 1514–1528.

    MathSciNet  Article  MATH  Google Scholar 

  9. [9]

    R. Diestel and S. Oum: Tangle-tree duality in abstract separation systems, arXiv:1701.02509, 2017.

    Google Scholar 

  10. [10]

    R. Diestel and S. Oum: Tangle-tree duality in graphs, matroids and beyond, arXiv:1701.02651, 2017.

    Google Scholar 

  11. [11]

    R. Diestel and G. Whittle: Tangles and the Mona Lisa, arXiv:1603.06652.

  12. [12]

    P. Eberenz: Characteristics of profiles, MSc dissertation, Hamburg 2015.

    Google Scholar 

  13. [13]

    A. Frank: Connections in Combinatorial Optimization, Oxford University Press, 2011.

    Google Scholar 

  14. [14]

    J. Geelen, B. Gerards, N. Robertson and G. Whittle: Obstructions to branch- decomposition of matroids, J. Combin. Theory (Series B) 96 (2006), 560–570.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    J. Geelen, B. Gerards and G. Whittle: Tangles, tree-decompositions and grids in matroids, J. Combin. Theory (Series B) 99 (2009), 657–667.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    R. E. Gomory and T. C. Hu: Multi-terminal network flows, J. Soc. Ind. Appl. Math. 9 (1961), 551–570.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    M. Grohe: Quasi-4-connected components, arXiv:1602.04505, 2016.

    Google Scholar 

  18. [18]

    F. Hundertmark: Profiles. An algebraic approach to combinatorial connectivity, https://archive.org/stream/arxiv-1110.6207/1110.6207, 2011.

    Google Scholar 

  19. [19]

    J. Oxley: Matroid Theory, Oxford University Press, 1992.

    Google Scholar 

  20. [20]

    N. Robertson and P. Seymour: Graph minors. X. Obstructions to tree- decomposition, J. Combin. Theory (Series B) 52 (1991), 153–190.

    MathSciNet  Article  MATH  Google Scholar 

  21. [21]

    W. T. Tutte: Graph Theory, Addison-Wesley, 1984.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Reinhard Diestel.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Diestel, R., Hundertmark, F. & Lemanczyk, S. Profiles of Separations: in Graphs, Matroids, and Beyond. Combinatorica 39, 37–75 (2019). https://doi.org/10.1007/s00493-017-3595-y

Download citation

Mathematics Subject Classification (2010)

  • 05C05
  • 05C40
  • 05C83
  • 05B35
  • 06A07