Shotgun assembly of unlabeled Erdos-Renyi graphs

Abstract

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 $(p_n)_{n=1}^\infty$ satisfy $n^{-1/2}\log^C n\leq p_n\leq c$. For each $n$, let $\Gamma_n$ be an unlabeled $G(n,p_n)$ Erd\"os-R\'enyi graph. Then with probability $1-o_n(1)$, any unlabeled graph $\tilde \Gamma_n$ on $n$ vertices with $\{N_{\tilde \Gamma_n}(v)\}_{v}=\{N_{\Gamma_n}(v)\}_{v}$ must coincide with $\Gamma_n$. This establishes $\tilde \Theta(n^{-1/2})$ as the transition range for the density parameter $p_n$ between reconstructability and non-reconstructability of Erd\"os-R\'enyi graphs from their $1$-neighborhoods, and resolves a problem of Gaudio and Mossel.