Spanning subgraphs of graphs partitioned into two isomorphic pieces

A graph has the neighbour-closed-co-neighbour, or ncc property, if for each of its vertices x, the subgraph induced by the neighbour set of x is isomorphic to the subgraph induced by the closed non-neighbour set of x. As proved by Bonato, Nowakowski [5], graphs with the ncc property are characterized by the existence of perfect matchings satisfying certain local conditions. In the present article, we investigate the spanning subgraphs of ncc graphs, which we name sub-ncc. Several equivalent characterizations of finite sub-ncc graphs are given, along with a polynomial time algorithm for their recognition. The infinite sub-ncc graphs are characterized, and we demonstrate the existence of a countable universal sub-ncc graph satisfying a strong symmetry condition called pseudo-homogeneity.