# Polynomial expansion and sublinear separators

## TL;DR

This paper proves that classes of graphs with sublinear separators have minors with bounded average degree, confirming a conjecture and linking separator properties to minor structure.

## Contribution

It establishes a relationship between sublinear separators and the average degree of depth-$k$ minors in graph classes, confirming a conjecture of Dvořák and Norin.

## Key findings

- Graphs with sublinear separators have minors with bounded average degree.
- The bound depends on the depth of the minor and the separator size.
- The result confirms a previously conjectured relationship in graph theory.

## Abstract

Let $\mathcal{C}$ be a class of graphs that is closed under taking subgraphs. We prove that if for some fixed $0<\delta\le 1$, every $n$-vertex graph of $\mathcal{C}$ has a balanced separator of order $O(n^{1-\delta})$, then any depth-$k$ minor (i.e. minor obtained by contracting disjoint subgraphs of radius at most $k$) of a graph in $\mathcal{C}$ has average degree $O\big((k \text{ polylog }k)^{1/\delta}\big)$. This confirms a conjecture of Dvo\v{r}\'ak and Norin.