## Abstract

Given a convex n-gon P in R{double-struck}2 with vertices in general position, it is well known that the simplicial complex θ(P) with vertex set given by diagonals in P and facets given by triangulations of P is the boundary complex of a polytope of dimension n-3. We prove that for any non-convex polygonal region P with n vertices and h+1 boundary components, θ(P) is a ball of dimension n+3h-4. We also provide a new proof that θ(P) is a sphere when P is convex with vertices in general position.

Original language | English |
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Pages (from-to) | 642-649 |

Number of pages | 8 |

Journal | Journal of Combinatorial Theory. Series A |

Volume | 117 |

Issue number | 6 |

DOIs | |

State | Published - Aug 2010 |

### Bibliographical note

Funding Information:E-mail addresses: [email protected] (B. Braun), [email protected] (R. Ehrenborg). URLs: http://www.ms.uky.edu/~braun (B. Braun), http://www.ms.uky.edu/~jrge (R. Ehrenborg). 1 Partially supported by National Science Foundation grant DMS-0758321. 2 Partially supported by National Security Agency grant H98230-06-1-0072.

### Funding

E-mail addresses: [email protected] (B. Braun), [email protected] (R. Ehrenborg). URLs: http://www.ms.uky.edu/~braun (B. Braun), http://www.ms.uky.edu/~jrge (R. Ehrenborg). 1 Partially supported by National Science Foundation grant DMS-0758321. 2 Partially supported by National Security Agency grant H98230-06-1-0072.

Funders | Funder number |
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National Science Foundation (NSF) | DMS-0758321 |

Directorate for Mathematical and Physical Sciences | 0758321 |

National Security Agency | H98230-06-1-0072 |

## Keywords

- Associahedra
- Discrete Morse theory
- Non-convex polygon
- Simplicial complex

## ASJC Scopus subject areas

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