Shotgun assembly of random graphs

TL;DR

This paper investigates the conditions under which Erdős-Rényi random graphs can be reconstructed from local neighborhood information, establishing thresholds and bounds for various neighborhood radii.

Contribution

It extends previous work by precisely determining reconstruction thresholds for larger neighborhoods and improving bounds for smaller neighborhoods in Erdős-Rényi graphs.

Findings

Reconstruction threshold for $r \\geq 3$ is established.

Improved polynomial bounds for $r=2$ case.

Sharpened results for $r=1$ neighborhood reconstruction.

Abstract

In the graph shotgun assembly problem, we are given the balls of radius $r$ around each vertex of a graph and asked to reconstruct the graph. We study the shotgun assembly of the Erd\H{o}s-R\'enyi random graph $\mathcal G(n,p)$ for a wide range of values of $r$. We determine the threshold for reconstructibility for each $r\geq 3$, extending and improving substantially on results of Mossel and Ross for $r=3$. For $r=2$, we give upper and lower bounds that improve on results of Gaudio and Mossel by polynomial factors. We also give a sharpening of a result of Huang and Tikhomirov for $r=1$.