Geodesic
Graph Exponentiation and Neighborhood Reconstruction
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 335-339

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Any graph G admits a neighborhood multiset 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 𝒩(G), that is, if G ≅ H whenever 𝒩(G) = 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.
Keywords: neighborhood reconstructible graphs, graph exponentiation 
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Hammack, Richard H. Graph Exponentiation and Neighborhood Reconstruction. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 1, pp. 335-339. http://geodesic.mathdoc.fr/item/DMGT_2021_41_1_a20/