Nowhere Dense Graph Classes and Dimension

Abstract

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 \(\mathcal{O}(n^\epsilon)\).

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Acknowledgements

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.

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Correspondence to Gwenaël Joret.

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Joret, G., Micek, P., Ossona de Mendez, P. et al. Nowhere Dense Graph Classes and Dimension. Combinatorica 39, 1055–1079 (2019). https://doi.org/10.1007/s00493-019-3892-8

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Mathematics Subject Classification (2010)

  • 06A07
  • 05C35