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 , where for . 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 , but not reconstructable for . Given the collection of distance-2 neighborhoods, G is exactly reconstructable for , but not reconstructable for .
E.M is partially supported by Vannevar Bush Faculty Fellowship ONR-N00014-20-1-2826, NSF awards CCF 1918421 and DMS-1737944, ARO MURI grant W911NF1910217 and by a Simons Investigator award.
We thank the anonymous reviewer for a very careful review. The reviewer’s comments caught several errors, and improved the clarity and presentation of this work.
"Shotgun assembly of Erdős-Rényi random graphs." Electron. Commun. Probab. 27 1 - 14, 2022. https://doi.org/10.1214/22-ECP445