Sparsity and dimension

TL;DR

This paper proves that posets of bounded height with cover graphs from classes with bounded expansion have bounded dimension, generalizing previous results and highlighting the limits of this property in more general graph classes.

Contribution

It establishes a new bounded dimension result for posets with cover graphs in classes with bounded expansion, extending prior work and identifying the boundaries of this property.

Findings

Posets of bounded height with cover graphs in bounded expansion classes have bounded dimension.

The result generalizes previous theorems for graphs excluding a fixed minor.

The bounded dimension property does not hold for nowhere dense classes or locally bounded treewidth graphs.

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 Ne\v{s}et\v{r}il 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.