On graphs isomorphic to their neighbour and non-neighbour sets

The paper contains a construction of a universal countable graph, different from the Rado graph, such that for any of its vertices both the neighbourhood and the non-neighbourhood induce subgraphs isomorphic to the whole graph. This solves an open problem proposed by A. Bonato; see Problem 20 in Cameron (2003) [5]. We supply a construction of several non-isomorphic graphs with the property, and consider tournaments with an analogous property.