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.
Categories: math.CO math.GT