Sparsity and Dimension

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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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