Skip to main content
SpringerLink
Log in

Sparsity and Dimension

  • Original Paper
  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We prove that posets of bounded height whose cover graphs belong to a fixed class with bounded expansion have bounded dimension. Bounded expansion, introduced by Nešetřil and Ossona de Mendez as a model for sparsity in graphs, is a property that is naturally satisfied by a wide range of graph classes, from graph structure theory (graphs excluding a minor or a topological minor) to graph drawing (e.g. graphs with bounded book thickness). Therefore, our theorem generalizes a number of results including the most recent one for posets of bounded height with cover graphs excluding a fixed graph as a topological minor. We also show that the result is in a sense best possible, as it does not extend to nowhere dense classes; in fact, it already fails for cover graphs with locally bounded treewidth.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Cs. Biró, M. T. Keller and S. J. Young: Posets with cover graph of pathwidth two have bounded dimension, Order 33 (2016), 195–212.

    Article  MathSciNet  MATH  Google Scholar 

  2. P. Chalermsook, B. Laekhanukit and D. Nanongkai: Graph products revisited: tight approximation hardness of induced matching, poset dimension and more, in: Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 1557–1576. SIAM, Philadelphia, PA, 2012.

    Google Scholar 

  3. V. Dujmović: Graph layouts via layered separators, J. Combin. Theory Ser. B 110 (2015), 79–89.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Dujmović and D. R. Wood: Stacks, queues and tracks: layouts of graph subdivisions, Discrete Math. Theor. Comput. Sci. 7 (2005), 155–201.

    MathSciNet  MATH  Google Scholar 

  5. B. Dushnik and E. W. Miller: Partially ordered sets, Amer. J. Math. 63 (1941), 600–610.

    Article  MathSciNet  MATH  Google Scholar 

  6. H. Enomoto, M. Sh. Miyauchi and K. Ota: Lower bounds for the number of edge-crossings over the spine in a topological book embedding of a graph, Discrete Appl. Math. 92 (1999), 149–155.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Eppstein: Diameter and treewidth in minor-closed graph families, Algorithmica 27 (2000), 275–291.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Felsner and W. T. Trotter: Dimension, graph and hypergraph coloring, Order 17 (2000), 167–177.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Felsner, W. T. Trotter and V. Wiechert: The Dimension of Posets with Planar Cover Graphs, Graphs Combin. 31 (2015), 927–939.

    Article  MathSciNet  MATH  Google Scholar 

  10. Z. Füredi and J. Kahn: On the dimensions of ordered sets of bounded degree, Order 3 (1986), 15–20.

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Grohe, S. Kreutzer and S. Siebertz: Characterisations of nowhere dense graphs, in: 33nd International Conference on Foundations of Software Technology and Theoretical Computer Science, volume 24 of LIPIcs. Leibniz Int. Proc. Inform., 21–40. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2013.

    Google Scholar 

  12. G. Joret, P. Micek, K. G. Milans, W. T. Trotter, B. Walczak and R. Wang: Tree-width and dimension, Combinatorica 36 (2016), 431–450.

    Article  MathSciNet  MATH  Google Scholar 

  13. G. Joret, P. Micek, W. T. Trotter, R. Wang and V. Wiechert: On the dimension of posets with cover graphs of treewidth 2, Order, to appear

  14. G. Joret, P. Micek and V. Wiechert: Sparsity and dimension, in: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’16, 1804–1813. SIAM, 2016.

    Google Scholar 

  15. D. Kelly: On the dimension of partially ordered sets, Discrete Math. 35 (1981), 135–156.

    Article  MathSciNet  MATH  Google Scholar 

  16. A. Lubotzky, R. Phillips and P. Sarnak: Ramanujan graphs, Combinatorica 8 (1988), 261–277.

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Micek and V. Wiechert: Topological minors of cover graphs and dimension, Submitted, arXiv:1504.07388.

  18. J. Nešetřil and P. O. de Mendez: Grad and classes with bounded expansion. I. Decompositions, European J. Combin. 29 (2008), 760–776.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Nešetřil and P. O. de Mendez: First order properties on nowhere dense structures, J. Symbolic Logic 75 (2010), 868–887.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Nešetřil and P. O. de Mendez: Sparsity, volume 28 of Algorithms and Combinatorics, Springer, Heidelberg, 2012, Graphs, structures, and algorithms.

    MATH  Google Scholar 

  21. J. Nešetřil, P. O. de Mendez and D. R. Wood: Characterisations and examples of graph classes with bounded expansion, European J. Combin. 33 (2012), 350–373.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Pach and G. Tóth: Graphs drawn with few crossings per edge, Combinatorica 17 (1997), 427–439.

    Article  MathSciNet  MATH  Google Scholar 

  23. F. Reidl, F. S. Villaamil and K. Stavropoulos: Characterising bounded expansion by neighbourhood complexity, arXiv:1603.09532.

  24. N. Streib and W. T. Trotter: Dimension and height for posets with planar cover graphs, European J. Combin. 35 (2014), 474–489.

    Article  MathSciNet  MATH  Google Scholar 

  25. W. T. Trotter: Combinatorics and partially ordered sets, Johns Hopkins Series in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1992, Dimension theory.

    MATH  Google Scholar 

  26. W. T. Trotter: Partially ordered sets, in: Handbook of combinatorics, Vol. 1, 2, 433–480. Elsevier Sci. B. V., Amsterdam, 1995.

    Google Scholar 

  27. W. T. Trotter, Jr. and J. I. Moore, Jr,: The dimension of planar posets, J. Combinatorial Theory Ser. B 22 (1977), 54–67.

    Article  MathSciNet  MATH  Google Scholar 

  28. B. Walczak: Minors and dimension, J. Combin. Theory Ser. B 122 (2017), 668–689.

    Article  MathSciNet  MATH  Google Scholar 

  29. M. Yannakakis: The complexity of the partial order dimension problem, SIAM J. Algebraic Discrete Methods 3 (1982), 351–358.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science Department, Université Libre de Bruxelles, Brussels, Belgium

    Gwenaël Joret

  2. Theoretical Computer Science Department Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland

    Piotr Micek

  3. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Piotr Micek & Veit Wiechert

Authors
  1. Gwenaël Joret
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr Micek
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Veit Wiechert
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gwenaël Joret.

Additional information

A preliminary version of this paper appeared as an extended abstract in the Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’16) [14].

G. Joret is supported by an ARC grant from the Wallonia-Brussels Federation of Belgium.

Piotr Micek was partially supported by the National Science Center of Poland under grant no. 2015/18/E/ST6/00299.

V. Wiechert is supported by the Deutsche Forschungsgemeinschaft within the research training group ‘Methods for Discrete Structures’ (GRK 1408).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Joret, G., Micek, P. & Wiechert, V. Sparsity and Dimension. Combinatorica 38, 1129–1148 (2018). https://doi.org/10.1007/s00493-017-3638-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-017-3638-4

Mathematics Subject Classification (2000)

Search

Navigation