Nowhere Dense Graph Classes and Dimension

TL;DR

This paper characterizes nowhere dense graph classes through poset dimension, showing that such classes have bounded poset dimension growth for certain posets, linking graph sparsity to order theory.

Contribution

It introduces a novel characterization of nowhere dense classes using poset dimension, connecting graph sparsity with order-theoretic properties.

Findings

Nowhere dense classes correspond to bounded poset dimension growth.

Poset dimension for certain posets in these classes is sublinear in size.

Provides a new perspective on graph sparsity via order theory.

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 $h \geq 1$ and every $\epsilon > 0$, posets of height at most $h$ with $n$ elements and whose cover graphs are in the class have dimension $\mathcal{O}(n^{\epsilon})$.