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Abstract Separation Systems

  • Reinhard Diestel
Article

Abstract

Abstract separation systems provide a simple general framework in which both tree-shape and high cohesion of many combinatorial structures can be expressed, and their duality proved. Applications range from tangle-type duality and tree structure theorems in graphs, matroids or CW-complexes to, potentially, image segmentation and cluster analysis. This paper is intended as a concise common reference for the basic definitions and facts about abstract separation systems in these and any future papers using this framework.

Keywords

Connectivity Graph Tangle Lattice Partial order Tree 

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References

  1. 1.
    Bowler, N., Diestel, R., Mazoit, F.: Tangle-tree duality in infinite graphs. In preparationGoogle Scholar
  2. 2.
    Carmesin, J., Diestel, R., Hundertmark, F., Stein, M.: Connectivity and tree structure in finite graphs. Combinatorica 34(1), 1–35 (2014)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Diestel, R.: Tree sets. arXiv:1512.03781. To appear in Order. doi:10.1007/s11083-017-9425-4 (2017)
  4. 4.
    Diestel, R.: Graph Theory (5th edition, 2016). Springer-Verlag, 2017. Electronic edition available at http://diestel-graph-theory.com/
  5. 5.
    Diestel, R., Erde, J.: Tangles in abstract separation systems. In preparationGoogle Scholar
  6. 6.
    Diestel, R., Erde, J., Eberenz, Ph.: Duality theorem for blocks and tangles in graphs. arXiv:1605.09139, to appear in SIAM J. Discrete Mathematics (2016)
  7. 7.
    Diestel, R., Hundertmark, F., Lemanczyk, S.: Profiles of separations: in graphs, matroids, and beyond. arXiv:1110.6207, to appear in Combinatorica
  8. 8.
    Diestel, R., Oum, S.: Tangle-tree duality in abstract separation systems. arXiv:1701.02509 (2017)
  9. 9.
    Diestel, R., Oum, S.: Tangle-tree duality in graphs, matroids and beyond. arXiv:1701.02651 (2017)
  10. 10.
    Diestel, R., Whittle, G.: Tangles and the Mona Lisa. arXiv:1603.06652 (2016)
  11. 11.
    Robertson, N., Seymour, P.D.: Graph Minors. X. Obstructions to tree-decomposition. J. Combin. Theory (Series B) 52, 153–190 (1991)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Reinhard Diestel
    • 1
  1. 1.Hamburg UniversityHamburgGermany

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