Shotgun assembly of Erdős-Rényi random graphs

Graph shotgun assembly refers to the problem of reconstructing a graph from collection local neighborhoods. In this paper, we consider Erdős–Rényi random graphs G(n,pn), where pn=n−α for 0<α<1. We both reconstruction up isomorphism as well exact (recovering vertex labels structure). show that given distance-1 neighborhoods, G is exactly reconstructable 0<α<1 3, but not 1 2<α<1. Given distance-2 α∈0,1 2∪1 2,3 5, 3 4<α<1.