Strongly sublinear separators and polynomial expansion

TL;DR

This paper proves a bidirectional relationship between strongly sublinear separators and polynomial expansion in hereditary graph classes, confirming a longstanding conjecture and deepening understanding of graph structure.

Contribution

It establishes the equivalence between strongly sublinear separators and polynomial expansion for hereditary graph classes, completing a key theoretical characterization.

Findings

Graphs with polynomial expansion have strongly sublinear separators.

Hereditary classes with strongly sublinear separators have polynomial expansion.

The result confirms a conjecture of the first author.

Abstract

A result of Plotkin, Rao, and Smith implies that graphs with polynomial expansion have strongly sublinear separators. We prove a converse of this result showing that hereditary classes of graphs with strongly sublinear separators have polynomial expansion. This confirms a conjecture of the first author.