Neighborhood structures and products of undirected graphs

Let G = ( V , E ) be a simple undirected graph. The neighborhood hypergraph N ( G ) = ( V , E N ) of G has the edge set E N = { e ⊆ V ∣ | e | ≥ 1 ∧ ∃ x ∈ V : e = N G ( x ) } . In a certain sense, this is a generalization of the well-known notion of the neighborhood graph N ( G ) = ( V , E N ) . For several products G 1 ∘ G 2 of simple undirected graphs G 1 and G 2 , we investigate the question how N ( G 1 ∘ G 2 ) / N ( G 1 ∘ G 2 ) can be constructed from G 1 , G 2 , N ( G 1 ) , N ( G 2 ) / N ( G 1 ) , N ( G 2 ) and vice versa.