The automorphism group of the circle equals the automorphism group of the intersection graph of all its chords, and the rational-chord subgraph is strongly universal and locally-complementation-invariant.
Graph Theory (6th edition)
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Circle graphs and the automorphism group of the circle
The automorphism group of the circle equals the automorphism group of the intersection graph of all its chords, and the rational-chord subgraph is strongly universal and locally-complementation-invariant.