We give an incremental polynomial time algorithm for enumerating the vertices of any polyhedron P(A, 1 ̄ ) = {x ∈ R | Ax ≥ 1 ̄ , x ≥ 0 ̄ }, when A is a totally unimodular matrix. Our algorithm is based on decomposing the hypergraph transversal problem for unimodular hypergraphs using Seymour’s decomposition of totally unimodular matrices, and may be of independent interest.

Drawing together techniques from combinatorics and computer science, we improve the census algorithm for enumerating closed minimal P-irreducible 3-manifold triangulations. In particular, new constraints are proven for face pairing graphs, and pruning techniques are improved using a modification of the union-find algorithm. Using these results we catalogue all 136 closed nonorientable P-irreduc...