Neighborhood hypergraphs of digraphs and some matrix permutation problems

Given a digraph D, the set of all pairs (N−(v), N(v)) constitutes the neighborhood dihypergraph N (D) of D. The Digraph Realization Problem asks whether a given dihypergraph H coincides with N (D) for some digraph D. This problem was introduced by Aigner and Triesch [2] as a natural generalization of the Open Neighborhood Realization Problem for undirected graphs, which is known to be NP-complete. We show that the Digraph Realization Problem remains NP-complete for orgraphs (orientations of undirected graphs). As a corollary, we show that the Matrix Skew-Symmetrization Problem for square {0, 1,−1} matrices (aij = −aji) is NP-complete. This result can be compared with the known fact that the Matrix Symmetrization Problem for square 0 − 1 matrices (aij = aji) is NP-complete. Extending a negative result of Fomin, Kratochv́ıl, Lokshtanov, Mancini, and Telle [15] we show that the Digraph Realization Problem remains NP-complete for almost all hereditary classes of digraphs defined by a unique minimal forbidden subdigraph. Finally, we consider the Matrix Complementation Problem for rectangular 0 − 1 matrices, and prove that it is polynomial-time equivalent to graph isomorphism. A related known result is that the Matrix Transposability Problem is polynomial-time equivalent to graph isomorphism. 2000 Mathematics Subject Classification: 68Q25 (Analysis of algorithms and problem complexity), 68R10 (Graph theory in relation to computer science), 05C62 (Graph representations), 05C20 (Directed graphs (digraphs), tournaments), 05C65 (Hypergraphs), 05C70 (Factorization, matching, covering and packing), 05C60 (Isomorphism problems), 05B20 ((0, 1)-matrices (combinatorics)), 05C85 (Graph algorithms). Neighborhood hypergraphs of digraphs and orgraphs, Graph Isomorphism Problem, matrix symmetrization, matrix complementation, symmetrizability, skew-symmetrizability, involutory automorphisms Acknowledgements: This research was partially supported by DIMACS, Center for Discrete Mathematics and Theoretical Computer Science, Rutgers University.