Discrete & Computational Geometry 52(3):427–449, 2014
(special issue of invited papers from the 29th Annual Symposium on Computational Geometry)
Proceedings of the 29th Annual Symposium on Computational Geometry, 37–46, 2013.
A topological quadrilateral mesh Q of a connected surface in R³ can be extended to a topological hexahedral mesh of the interior domain Ω if and only if Q has an even number of quadrilaterals and no odd cycle in Q bounds a surface inside Ω. Moreover, if such a mesh exists, the required number of hexahedra is within a constant factor of the minimum number of tetrahedra in a triangulation of Ω that respects Q. Finally, if Q is given as a polyhedron in R³ with quadrilateral facets, a topological hexahedral mesh of the polyhedron can be constructed in polynomial time if such a mesh exists. All our results extend to domains with disconnected boundaries. Our results naturally generalize results of Thurston, Mitchell, and Eppstein for genus-zero and bipartite meshes, for which the odd-cycle criterion is trivial.