## Abstract

A topological quadrilateral mesh Q of a connected surface in (Formula Prseented.) can be extended to a topological hexahedral mesh of the interior domain (Formula Prseented.) if and only if Q has an even number of quadrilaterals and no odd cycle in Q bounds a surface inside (Formula Prseented.). 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 (Formula Prseented.) that respects Q. Finally, if Q is given as a polyhedron in (Formula Prseented.) 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.

Original language | English (US) |
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Pages (from-to) | 427-449 |

Number of pages | 23 |

Journal | Discrete and Computational Geometry |

Volume | 52 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 2014 |

## Keywords

- Computational topology
- Cube complexes
- Homology
- Mesh generation

## ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics