Shotgun Assembly of Erdos-Renyi Random Graphs

Graph shotgun assembly refers to the problem of reconstructing a graph from a collection of local neighborhoods. In this paper, we consider shotgun assembly of ErdH{o}s-Renyi random graphs $G(n, p_n)$, where $p_n = n^{-alpha}$ for $0 < alpha < 1$. We consider both reconstruction up to isomorphism as well as exact reconstruction (recovering the vertex labels as well as the structure). We show that given the collection of distance-$1$ neighborhoods, $G$ is exactly reconstructable for $0 < alpha < frac{1}{3}$, but not reconstructable for $frac{1}{2} < alpha < 1$. Given the collection of distance-$2$ neighborhoods, $G$ is exactly reconstructable for $0 < alpha < frac{1}{2}$, but not reconstructable for $frac{3}{4} < alpha < 1$.