Nowhere Dense Graph Classes and Dimension


Nowhere dense graph classes provide one of the least restrictive notions of sparsity for graphs. Several equivalent characterizations of nowhere dense classes have been obtained over the years, using a wide range of combinatorial objects. In this paper we establish a new characterization of nowhere dense classes, in terms of poset dimension: A monotone graph class is nowhere dense if and only if for every h⩾1 and every ε > 0, posets of height at most h with n elements and whose cover graphs are in the class have dimension \(\mathcal{O}(n^\epsilon)\).

This is a preview of subscription content, access via your institution.


  1. [1]

    H. Adler and I. Adler: Interpreting nowhere dense graph classes as a classical notion of model theory, European Journal of Combinatorics36 (2014), 322–330.

    MathSciNet  Article  Google Scholar 

  2. [2]

    R. Diestel: Graph theory, volume 173 of Graduate Texts in Mathematics, Springer, Heidelberg, fourth edition, 2010.

    Google Scholar 

  3. [3]

    K. Eickmeyer, A. C. Giannopoulou, S. Kreutzer, O. Kwon, M. Pilipczuk, R. Rabinovich and S. Siebertz: Neighborhood complexity and kernelization for nowhere dense classes of graphs, in: Ioannis Chatzigiannakis, Piotr Indyk, Fabian Kuhn, and Anca Muscholl, editors, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017), volume 80 of Leibniz International Proceedings in Informatics (LIPIcs), 1–14, Dagstuhl, Germany, 2017. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

    Google Scholar 

  4. [4]

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

    MathSciNet  Article  Google Scholar 

  5. [5]

    M. Grohe, S. Kreutzer, R. Rabinovich, S. Siebertz and K. Stavropoulos: Colouring and covering nowhere dense graphs, in: Graph-Theoretic Concepts in Computer Science 41st International Workshop, WG 2015, 325–338, 2015.

    Google Scholar 

  6. [6]

    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 

  7. [7]

    M. Grohe, S. Kreutzer and S. Siebertz: Deciding first-order properties of nowhere dense graphs, J. ACM64 (2017), 1–32.

    MathSciNet  Article  Google Scholar 

  8. [8]

    J. van den Heuvel, P. Ossona de Mendez, D. Quiroz, R. Rabinovich and S. Siebertz: On the generalised colouring numbers of graphs that exclude a fixed minor, European Journal of Combinatorics66 (Supplement C) (2017), 129–144.

    MathSciNet  Article  Google Scholar 

  9. [9]

    G. Joret, P. Micek, K. G. Milans, W. T. Trotter, B. Walczak and R. Wang: Tree-width and dimension, Combinatorica36 (2016), 431–450.

    MathSciNet  Article  Google Scholar 

  10. [10]

    G. Joret, P. Micek, W. T. Trotter, R. Wang and V. Wiechert: On the dimension of posets with cover graphs of treewidth 2, Order34 (2017), 185–234. arXiv:1406.3397.

    MathSciNet  Article  Google Scholar 

  11. [11]

    G. Joret, P. Micek and V. Wiechert: Planar posets have dimension at most linear in their height, SIAM J. Discrete Math.31 (2018), 2754–2790.

    MathSciNet  Article  Google Scholar 

  12. [12]

    G. Joret, P. Micek and V. Wiechert: Sparsity and dimension, Combinatorica38 (2018), 1129–1148.

    MathSciNet  Article  Google Scholar 

  13. [13]

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

    MathSciNet  Article  Google Scholar 

  14. [14]

    H. A. Kierstead and D. Yang: Orderings on graphs and game coloring number, Order20 (2003), 255–264.

    MathSciNet  Article  Google Scholar 

  15. [15]

    S. Kreutzer, M. Pilipczuk, R. Rabinovich and S. Siebertz: The generalised colouring numbers on classes of bounded expansion, in: Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, editors, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016), volume 58 of Leibniz International Proceedings in Informatics (LIPIcs), 1–13, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.

    Google Scholar 

  16. [16]

    S. Kreutzer, R. Rabinovich and S. Siebertz: Polynomial kernels and wideness properties of nowhere dense graph classes, in: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '17, 1533–1545, 2017.

    Google Scholar 

  17. [17]

    P. Micek and V. Wiechert: Topological minors of cover graphs and dimension, Journal of Graph Theory86 (2017), 295–314.

    MathSciNet  Article  Google Scholar 

  18. [18]

    J. Nešsetrřil and P. P. Ossona de Mendez: Existence of modeling limits for sequences of sparse structures, arXiv:1608.00146.

  19. [19]

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

    MathSciNet  Article  Google Scholar 

  20. [20]

    J. Nešsetrřil and P. Ossona de Mendez: First order properties on nowhere dense structures, J. Symbolic Logic75 (2010), 868–887.

    MathSciNet  Article  Google Scholar 

  21. [21]

    J. Nešsetrřil and P. Ossona de Mendez: How many F's are there in G? European Journal of Combinatorics32 (2011), 1126–1141.

    MathSciNet  Article  Google Scholar 

  22. [22]

    J. Nešsetrřil and P. Ossona de Mendez: On nowhere dense graphs, European J. Combin.32 (2011), 600–617.

    MathSciNet  Article  Google Scholar 

  23. [23]

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

    Google Scholar 

  24. [24]

    J. Nešsetrřil and P. Ossona de Mendez: Structural sparsity, Uspekhi Matematich-eskikh Nauk71 (2016), 85–116. (Russian Math. Surveys 71 79-107).

    MathSciNet  Article  Google Scholar 

  25. [25]

    M. Pilipczuk, S. Siebertz and Sz. Toruńczyk: On wideness and stability, arXiv:1705.09336.

  26. [26]

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

    MathSciNet  Article  Google Scholar 

  27. [27]

    W. T. Trotter, Jr. and J. I. Moore, Jr: The dimension of planar posets, J. Combinatorial Theory Ser. B22 (1977), 54–67.

    MathSciNet  Article  Google Scholar 

  28. [28]

    B. Walczak: Minors and dimension, J. Combin. Theory Ser. B122 (2017), 668–689. Extended abstract in Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '15). arXiv:1407.4066.

    MathSciNet  Article  Google Scholar 

  29. [29]

    V. Wiechert: Cover graphs and order dimension, PhD thesis, TU Berlin, 2018.

  30. [30]

    X. Zhu: Colouring graphs with bounded generalized colouring number. Discrete Math.309 (2009), 5562–5568.

    MathSciNet  Article  Google Scholar 

Download references


We are much grateful to the anonymous referees for their very helpful comments. In particular, we thank one referee for pointing out an error in the proof of Claim 10 regarding how element q was chosen, and another referee for her/his many suggestions on how to improve the exposition of the proofs and shorthen the arguments.

Author information



Corresponding author

Correspondence to Gwenaël Joret.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Joret, G., Micek, P., Ossona de Mendez, P. et al. Nowhere Dense Graph Classes and Dimension. Combinatorica 39, 1055–1079 (2019).

Download citation

Mathematics Subject Classification (2010)

  • 06A07
  • 05C35