Shotgun Assembly of Labeled Graphs

We consider the problem of reconstructing graphs or labeled graphs from neighborhoods of a given radius r. Special instances of this problem include the well-known: DNA shotgun assembly; the lesser-known: neural network reconstruction; and a new problem: assembling random jigsaw puzzles. We provide some necessary and some sufficient conditions for correct recovery both in combinatorial terms and for some generative models including random labelings of lattices, Erdös-Rényi random graphs, and a random jigsaw puzzle model. Many open problems and conjectures are provided.