A proof of a conjecture of Sabidussi on graphs idempotent under the lexicographic product

In 1960, Sabidussi conjectured that if a graph G is isomorphic to the lexicographic product G [ G ] , then the wreath product of Aut ( G ) by itself is a proper subgroup of Aut ( G [ G ] ) . A positive answer is provided by constructing an automorphism Ψ of G [ G ] which satisfies: for every vertex x of G , there is an infinite subset I ( x ) of V ( G ) such that Ψ ( { x } × V ( G ) ) = I ( x ) × V ( G ) .