Graph Exponentiation and Neighborhood Reconstruction

Any graph $G$ admits a neighborhood multiset \(\mathscr{N}(G) = \{N_G(x) | x ∈ V (G)\}\) whose elements are precisely the open neighborhoods of $G$. We say $G$ is neighborhood reconstructible if it can be reconstructed from \(\mathscr{N}(G)\), that is, if \(G ≅ H\) whenever \(\mathscr{N}(G) = \mathscr{N}(H)\) for some other graph $H$. This note characterizes neighborhood reconstructible graphs as those graphs $G$ that obey the exponential cancellation \(G^{K_2} ≅ H^{K_2} ⇒ G ≅ H\).