# Extending partial isometries of antipodal graphs

## TL;DR

This paper establishes the extension property for partial automorphisms (EPPA) for all antipodal classes in Cherlin's list of metrically homogeneous graphs, using a novel method that overcomes previous limitations.

## Contribution

It introduces a new general technique for proving EPPA that bypasses the need for automorphism-preserving completions, applied to antipodal classes.

## Key findings

- Proves EPPA for all antipodal classes in Cherlin's list.
- Develops a new method combining recent theorems and ideas from previous EPPA proofs.
- Answers a question posed by Aranda et al.

## Abstract

We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin's list of metrically homogeneous graphs, thereby answering a question of Aranda et al. This paper should be seen as the first application of a new general method for proving EPPA which can bypass the lack of an automorphism-preserving completion. It is done by combining the recent strengthening of the Herwig--Lascar theorem by Hubi\v{c}ka, Ne\v{s}et\v{r}il and the author with the ideas of the proof of EPPA for two-graphs by Evans et al.