Reconstructing a Graph from its Neighborhood Lists

Associate to a finite labeled graph G ( V, E ) its multiset of neighborhoods ( G ) = {N(υ): υ ∈ V }. We discuss the question of when a list is realizable by a graph, and to what extent G is determined by ( G ). The main results are: the decision problem is NP-complete; for bipartite graphs the decision problem is polynomially equivalent to Graph Isomorphism; forests G are determined up to isomorphism by ( G ); and if G is connected bipartite and ( H ) = ( G ), then H is completely described.