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 $r=1$.

翻译：图形散弹枪组装是指从每个顶端周围收集的美元球体中重建一个图的问题。 我们研究的是这个问题,用于Erd\H{o}s-R\'enyi随机图$G\in\mathcal G(n,p)$,以及各种值$r. 我们确定3美元(r\geq)的可重建性准确阈值,它改进和概括了Mossel和Ross的产值为$r=3。 此外,我们还在2-可重建性阈值上给出了更好的上下限,用多种系数改进了Gaudio和Mossel的产值。 我们还对黄蜂和Tikhomirorov的产值为$1美元的结果给出了更高的下限。