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Tree Sets

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Abstract

We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and matroids etc.

Unlike graph-theoretical or order trees, these tree sets can provide a suitable formalization of tree structure also for infinite graphs, matroids, and set partitions. Order trees reappear as oriented tree sets.

We show how each of the above structures defines a tree set, and which additional information, if any, is needed to reconstruct it from this tree set.

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References

  1. Bowler, N., Diestel, R., Mazoit, F.: Tangle-tree duality in infinite graphs. In preparation

  2. Diestel, R.: Abstract separation systems, arXiv:1406.3797. Order (2017). doi:10.1007/s11083-017-9424-5

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

  4. Diestel, R., Erde, J., Eberenz, P.: Duality theorem for blocks and tangles in graphs. arXiv:1605.09139, to appear in SIAM J. Discrete Mathematics (2016)

  5. Diestel, R., Hundertmark, F., Lemanczyk, S.: Profiles of separations: in graphs, matroids, and beyond. arXiv:1110.6207, to appear in Combinatorica (2017)

  6. Diestel, R., Kneip, J.: Profinite tree sets. In preparation

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

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

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

  10. Dunwoody, M.J.: Inaccessible groups and protrees. J. Pure Appl. Algebra 88, 63–78 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Dunwoody, M.J.: Groups acting on protrees. J. Lond. Math. Soc. 56(2), 125–136 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gollin, P., Kneip, J.: Representations of infinite tree sets. In preparation

  13. Hundertmark, F.: Profiles. An algebraic approach to combinatorial connectivity. arXiv:1110.6207 (2011)

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Woess, W.: Graphs and groups with tree-like properties. J. Comb. Theory Ser. B 47, 361–371 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Diestel, R. Tree Sets. Order 35, 171–192 (2018). https://doi.org/10.1007/s11083-017-9425-4

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  • DOI: https://doi.org/10.1007/s11083-017-9425-4

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