Neighborhood structures and products of undirected graphs

Let G=(V,E)G=(V,E) be a simple undirected graph. The neighborhood hypergraph  N(G)=(V,EN)N(G)=(V,EN) of GG has the edge set EN={e⊆V∣|e|≥1∧∃x∈V:e=NG(x)}. In a certain sense, this is a generalization of the well-known notion of the neighborhood graph  N(G)=(V,EN)N(G)=(V,EN). For several products G1∘G2G1∘G2 of simple undirected graphs G1G1 and G2G2, we investigate the question how N(G1∘G2)/N(G1∘G2)N(G1∘G2)/N(G1∘G2) can be constructed from G1G1, G2G2, N(G1)N(G1), N(G2)/N(G1)N(G2)/N(G1), N(G2)N(G2) and vice versa.