On automorphisms of the countable p.e.c. graph

The (countable) perturbed existentially closed graph S (Gordinowicz, 2010 [5]) was introduced by the second author as a solution to a problem stated by Bonato (Problem 20 in Cameron (2003) [3]). The graph S is not isomorphic to the Rado graph, nevertheless it has the NNc property in the sense that subgraphs induced by the neighbourhood and by the non-neighbourhood of each vertex of S are isomorphic to S. The graph S is given explicitly and is also uniquely-up to an isomorphism-characterized by a perturbed existential closure property (Gordinowicz, 2010 [5]). In the paper we characterize isomorphisms of finite, induced subgraphs of S which can be extended to global automorphisms.