# Embeddability of arrangements of pseudocircles and graphs on surfaces

## TL;DR

This paper establishes a precise criterion for when arrangements of pseudocircles can be embedded into orientable surfaces of a given genus, extending previous results from the sphere to higher-genus surfaces.

## Contribution

It proves that embeddability into a genus g surface depends on all subarrangements of size at most 4g+4, and shows this bound is optimal, generalizing to arrangements of graphs.

## Key findings

- Embeddability into genus g surfaces characterized by subarrangements of size 4g+4.
- Bound for subarrangement size is tight and cannot be improved.
- Results extend to arrangements of graphs beyond pseudocircles.

## Abstract

A pseudocircle is a simple closed curve on some surface; an arrangement of pseudocircles is a collection of pseudocircles that pairwise intersect in exactly two points, at which they cross. Ortner proved that an arrangement of pseudocircles is embeddable into the sphere if and only if all of its subarrangements of size at most four are embeddable into the sphere, and asked if an analogous result holds for embeddability into orientable surfaces of higher genus. We answer this question positively: An arrangement of pseudocircles is embeddable into an orientable surface of genus~$g$ if and only if all of its subarrangements of size at most $4g+4$ are. Moreover, this bound is tight. We actually have similar results for a much general notion of arrangement, which we call an \emph{arrangement of graphs}.