Circle graphs and the automorphism group of the circle

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.