Circle graphs and the automorphism group of the circle

TL;DR

This paper establishes that the automorphism group of the circle graph formed by chords on a circle is identical to the automorphism group of the circle itself, revealing deep symmetry properties and universality features.

Contribution

It proves the automorphism group of the circle graph matches that of the circle and introduces a strongly universal rational chord subgraph invariant under local complementation.

Findings

Aut({}^1) equals automorphism group of the circle graph.

The rational chord subgraph is strongly universal and invariant under local complementation.

Only K_2 and the Rado graph share similar invariance properties.

Abstract

We prove that $Aut({\mathbb S}^1)$ coincides with the automorphism group of the \emph{circle graph} $\mathcal{C}$, i.e. the intersection graph of the family of chords of ${\mathbb S}^1$. We prove that the countable subgraph of $\mathcal{C}$ induced by the rational chords is a strongly universal element of the family of circle graphs, and that it is invariant under local complementation. The only other known connected graphs that have the latter property are $K_2$ and the Rado graph.