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

In 1960, Sabidussi conjectured that if a graph GG is isomorphic to the lexicographic product G[G]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 ΨΨ of G[G]G[G] which satisfies: for every vertex xx of GG, there is an infinite subset I(x)I(x) of V(G)V(G) such that Ψ({x}×V(G))=I(x)×V(G)Ψ({x}×V(G))=I(x)×V(G).