Shotgun assembly of random graphs

11/25/2022

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by   Tom Johnston, et al.

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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 Erdős-Rényi random graph G∈𝒢(n,p), and for a wide range of values of r. We determine the exact thresholds for r-reconstructibility for r≥ 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.