Partitioning a graph into two isomorphic pieces

A simple graph G has the neighbour-closed-co-neighbour property, or ncc property, 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. We present characterizations of graphs with the ncc property via the existence of certain perfect matchings, and thereby prove that the ncc graphs can be recognized in polynomial time. Operations preserving the ncc property are discussed, and the graphs with the ncc property that are r-regular, where r ≤ 5, are classified. The graphs with the ncc property that are locally H, for certain graphs H are classified. Infinite graphs with the ncc property are discussed.