Neighborhood hypergraphs of bipartite graphs

Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity status and their relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular we recall results from the literature showing that matrix symmetrization is in fact NP-hard, while graph isomorphism is still an open problem. One of the equivalent formulations of matrix symmetrization is the problem of recognizing if a given hypergraph can be realized as the neighborhood hypergraph of a graph. While all variants of this problem are NP-hard, in general, the restriction of one of the variants for bipartite graphs is known to be equivalent with graph isomorphism. Generalizing these results, we consider several other variants of the bipartite neighborhood recognition problem, and show that all these variants are either polynomial-time solvable, or are equivalent with graph isomorphism. Finally, we consider the uniqueness of bipartite neighborhood realizations of hypergraphs. In general, a realization, if exists, may not be unique for any variants of the problem, even when realizations are restricted to bipartite graphs. Settling an open problem, we prove uniqueness of bipartite realizations for two variants, for the so called open and closed neighborhood hypergraphs of bipartite graphs. 2000 Mathematics Subject Classification: 68Q25 (Analysis of algorithms and problem complexity), 68R10 (Graph theory in relation to computer science), 05C62 (Graph representations), 05C65 (Hypergraphs), 05C70 (Factorization, matching, covering and packing), 05C60 (Isomorphism problems), 05C85 (Graph algorithms).