Sublinear separators, fragility and subexponential expansion

TL;DR

This paper establishes that certain graph classes with small balanced separators exhibit subexponential expansion, linking separator size to graph expansion properties.

Contribution

It proves a partial converse relating small separators to subexponential expansion in bounded degree, subgraph-closed graph classes, introducing new graph decomposition techniques.

Findings

Graphs with polynomially smaller balanced separators have subexponential expansion.

Introduces new graph decomposition methods for analyzing graph structure.

Provides a partial converse to a known result by Nesetril and Ossona de Mendez.

Abstract

Let G be a subgraph-closed graph class with bounded maximum degree. We show that if G has balanced separators whose size is smaller than linear by a polynomial factor, then G has subexponential expansion. This gives a partial converse to a result of Ne\v{s}et\v{r}il and Ossona de Mendez. As an intermediate step, the proof uses a new kind of graph decompositions.