Partitioning a graph into two pieces, each isomorphic to the other or to its complement

A simple graph G has the generalized-neighbour-closed-co-neighbour property, or is a gncc graph, if for all vertices x of G, the subgraph, induced by the set of neighbours of x, is isomorphic to the subgraph, induced by the set of non-neighbours of x, or is isomorphic to its complement. If every vertex x satisfies the first condition (that is, the subgraphs, induced by its set of neighbours, and by its set of non-neighbours, are isomorphic), then the graph has the neighbour-closed-co-neighbour property, or is an ncc graph. In [A. Bonato, R. Nowakowski, Partitioning a graph into two isomorphic pieces, J. Graph Theory, 44 (2003) 1–14], the ncc graphs were characterized and a polynomial time algorithm was given for their recognition. In this paper we show that all gncc graphs are also ncc, that is, we prove that the two families of graphs, defined above, are identical. Finally, we present some of the properties of an interesting family of graphs, that is derived from the proof of the claim above, and we give a polynomial time algorithm to recognize such graphs.