The deck of a graph is the multiset consisting of all unlabelled subgraphs obtained from by deleting a vertex in all possible ways (counted according to multiplicity).
See Wikipedia's Reconstruction Conjecture for more on this problem.
*[K] P. J. Kelly, A congruence theorem for trees, Pacific J. Math., 7 (1957), 961–968.
*[U] S. M. Ulam, A collection of mathematical problems, Wiley, New York, 1960.
* indicates original appearance(s) of problem.