Sparsity and Dimension

Combinatorica volume 38, pages1129–1148(2018)Cite this article

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.

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Author information

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
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  2. Piotr Micek
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  3. Veit Wiechert
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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).

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

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Mathematics Subject Classification (2000)

  • 06A07
  • 05C35