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arXiv · 1507.01120

Sparsity and dimension

Gwenaël Joret·Piotr Micek·Veit Wiechert
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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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Gwenaël Joret, Piotr Micek, Veit Wiechert. 2017-01-09. Sparsity and dimension. https://doi.org/10.1007/s00493-017-3638-4

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