Advertisement

Combinatorica

, Volume 39, Issue 4, pp 879–910 | Cite as

Tangle-Tree Duality: In Graphs, Matroids and Beyond

  • Reinhard Diestel
  • Sang-il OumEmail author
Article

Abstract

We apply a recent tangle-tree duality theorem in abstract separation systems to derive tangle-tree-type duality theorems for width-parameters in graphs and matroids.We further derive a duality theorem for the existence of clusters in large data sets.

Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a tangle-type duality theorem for tree-width.

Our results can also be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width.

Mathematics Subject Classification (2010)

05C40 05C75 05C83 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    O. Amini, F. Mazoit, N. Nisse and S. Thomasse: Submodular partition functions, Discrete Appl. Math. 309 (2009), 6000–6008.MathSciNetzbMATHGoogle Scholar
  2. [2]
    D. Bienstock, N. Robertson, P. Seymour and R. Thomas: Quickly excluding a forest, J. Combin. Theory Ser. B 52 (1991), 274–283.MathSciNetCrossRefGoogle Scholar
  3. [3]
    N. Bowler: Presentation at Hamburg workshop on graphs and matroids, Spiekeroog 2014.Google Scholar
  4. [4]
    J. Carmesin, R. Diestel, F. Hundertmark and M. Stein: Connectivity and tree structure in finite graphs, Combinatorial 34 (2014), 1–35.MathSciNetCrossRefGoogle Scholar
  5. [5]
    R. Diestel: Graph Theory, Springer, 4th edition, 2010.CrossRefGoogle Scholar
  6. [6]
    R. Diestel: Graph Theory (5th edition), Springer-Verlag, 2017, Electronic edition available at http://diestel-graph-theory.com/.CrossRefGoogle Scholar
  7. [7]
    R. Diestel: Abstract separation systems, Order 35 (2018), 157–170.MathSciNetCrossRefGoogle Scholar
  8. [8]
    R. Diestel, P. Eberenz and J. Erde: Duality theorem for blocks and tangles in graphs, SIAM J. Discrete Math. 31 (2017), 1514–1528.MathSciNetCrossRefGoogle Scholar
  9. [9]
    R. Diestel and S. Oum: Unifying duality theorems for width parameters, II. General duality, arXiv:1406.3798, 2014.zbMATHGoogle Scholar
  10. [10]
    R. Diestel and S. Oum: Tangle-tree duality in abstract separation systems, arXiv:1701.02509, 2017.Google Scholar
  11. [11]
    R. Diestel and S. Oum: Tangle-tree duality in graphs, matroids and beyond (extended version), arXiv:1701.02651, 2017.Google Scholar
  12. [12]
    R. Diestel and G. Whittle: Tangles and the Mona Lisa, arXiv:1603.06652.Google Scholar
  13. [13]
    J. Geelen, B. Gerards, N. Robertson and G. Whittle: Obstructions to branch-decomposition of matroids, J. Combin. Theory (Series B) 96 (2006), 560–570.MathSciNetCrossRefGoogle Scholar
  14. [14]
    P. Hlineny and G. Whittle: Matroid tree-width, European J. Combin. 27 (2006), 1117–1128.MathSciNetCrossRefGoogle Scholar
  15. [15]
    P. Hlineny and G. Whittle: Addendum to matroid tree-width, European J. Combin. 30 (2009), 1036–1044.MathSciNetCrossRefGoogle Scholar
  16. [16]
    C.-H. Liu: Packing and covering immersions in 4-edge-connected graphs, arXiv:1505.00867, 2015.Google Scholar
  17. [17]
    L. Lyaudet, F. Mazoit and S. Thomasse: Partitions versus sets: a case of duality, European J. Combin. 31 (2010), 681–687.MathSciNetCrossRefGoogle Scholar
  18. [18]
    S. Oum and P. Seymour: Approximating clique-width and branch-width, J. Combin. Theory Ser. B 96 (2006), 514–528.MathSciNetCrossRefGoogle Scholar
  19. [19]
    J. Oxley: Matroid Theory, Oxford University Press, 1992.zbMATHGoogle Scholar
  20. [20]
    N. Robertson and P. Seymour: Graph minors. X. Obstructions to tree-decomposition, J. Combin. Theory (Series B) 52 (1991), 153–190.MathSciNetCrossRefGoogle Scholar
  21. [21]
    P. Seymour and R. Thomas: Graph searching and a min-max theorem for tree-width, J. Combin. Theory (Series B) 58 (1993), 22–33.MathSciNetCrossRefGoogle Scholar
  22. [22]
    P. Seymour and R. Thomas: Call routing and the ratcatcher, Combinatorica 14 (1994), 217–241.MathSciNetCrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2019

Authors and Affiliations

  1. 1.Mathematisches SeminarUniversität HamburgHamburgGermany
  2. 2.Discrete Mathematics GroupInstitute for Basic Science (IBS)DaejeonKorea
  3. 3.Department of Mathematical SciencesKAISTDaejeonKorea

Personalised recommendations