Shotgun assembly of Erdős-Rényi random graphs

Abstract

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 Erdős–Rényi random graphs $G(n,p_n)$, where $p_n=n^{-\mathit{\alpha }}$ for $0<\mathit{\alpha }<1$. 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 $0<\mathit{\alpha }<\frac{1}{3}$, but not reconstructable for $\frac{1}{2}<\mathit{\alpha }<1$. Given the collection of distance-2 neighborhoods, G is exactly reconstructable for $\mathit{\alpha }\in \left(0,\frac{1}{2}\right)\cup \left(\frac{1}{2},\frac{3}{5}\right)$, but not reconstructable for $\frac{3}{4}<\mathit{\alpha }<1$.