Classification of connected countably infinite graphs invariant under local complementation

Determine whether there exists a non-empty, connected, countably infinite vertex-transitive graph invariant under local complementation that is not isomorphic to the Rado graph or the complete graph on two vertices, and determine whether infinitely many such graphs exist; in particular, determine whether the rational circle graph is the unique non-empty, connected, countable circle graph with this invariance.

Background

The paper proves that the rational circle graph is strongly universal for countable circle graphs and invariant under local complementation. It also observes that, among connected graphs known to the author, only the Rado graph and K_2 share this invariance. The stated problem asks whether additional examples exist and whether the rational circle graph is uniquely characterized by this property within countable circle graphs.

References

Is there a non-empty, connected, countably infinite (vertex-transitive) graph which is invariant under local complementation and is not isomorphic to the Rado graph or $$? Are there infinitely many such graphs? In particular, is $$ the unique, up to isomorphism, non-empty, connected, countable circle graph which is invariant under local complementation? It could also be interesting to look for finite (circle) graphs that are invariant under local complementation. I do not know of any connected example with more than one edge.

— Circle graphs and the automorphism group of the circle  (2501.07698 - Georgakopoulos, 13 Jan 2025) in Problem following Theorem \ref{thm univ}, Introduction