# On tope graphs of complexes of oriented matroids

## TL;DR

This paper provides new graph-theoretic characterizations of tope graphs of complexes of oriented matroids, including algorithms for recognition and implications for related graph classes, solving longstanding open problems.

## Contribution

It introduces two novel characterizations of tope graphs using excluded minors and gated antipodal subgraphs, and offers polynomial recognition algorithms.

## Key findings

- Characterization via excluded partial cube minors
- Recognition algorithms for tope graphs
- Confirmation that all finite Pasch graphs are tope graphs

## Abstract

We give two graph theoretical characterizations of tope graphs of (complexes of) oriented matroids. The first is in terms of excluded partial cube minors, the second is that all antipodal subgraphs are gated. A direct consequence is a third characterization in terms of zone graphs of tope graphs. Further corollaries include a characterization of topes of oriented matroids due to da Silva, another one of Handa, a characterization of lopsided systems due to Lawrence, and an intrinsic characterization of tope graphs of affine oriented matroids. Furthermore, we obtain polynomial time recognition algorithms for tope graphs of the above and a finite list of excluded partial cube minors for the bounded rank case. In particular, this answers a relatively long-standing open question in oriented matroids. Another consequence is that all finite Pasch graphs are tope graphs of complexes of oriented matroids, which confirms a conjecture of Chepoi and the two authors.