Shotgun assembly of unlabeled Erdos-Renyi graphs

Given a positive integer n, an unlabeled graph G on n vertices, and a vertex v of G, let N-G(v) be the subgraph of G induced by vertices of G of distance at most one from v. We show that there are universal constants C, c > 0 with the following property. Let the sequence (p(n))(n=1)(8) satisfy n(-1/2) log(n)(C) <= pn <= c. For each n, let Gamma(n) be an unlabeled G(n, p(n)) Erdos-Renyi graph. Then with probability 1 - o(n)(1), any unlabeled graph (Gamma) over tilde (n) on n vertices with {N (Gamma) over tilde (n) (v)}(v) = {N-Gamma n (v)}(v) must coincide with Gamma(n). This establishes (Theta) over tilde (n(-1/2)) as the transition range for the density parameter pn between reconstructability and non-reconstructability of Erdos-Renyi graphs from their 1-neighborhoods, and resolves a problem of Gaudio and Mossel from (Electron Commun Probab 27: 1-14, 2022)