Shotgun assembly of random graphs

Graph shotgun assembly refers to the problem of reconstructing a graph from the collection of $r$-balls around each vertex. We study this problem for an Erd\H{o}s-R\'enyi random graph $G\in \mathcal G(n,p)$, and for a wide range of values of $r$. We determine the exact thresholds for $r$-reconstructibility for $r\geq 3$, which improves and generalises the result of Mossel and Ross for $r=3$. In addition, we give better upper and lower bounds on the threshold of 2-reconstructibility, improving the results of Gaudio and Mossel by polynomial factors. We also give an improved lower bound for the result of Huang and Tikhomirov for $r=1$.