### Abstract

Let P be a simple polygon with n vertices. The dual graph T^{⁎} of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n^{3}logn) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2⋅⌈log_{2}(n/3)⌉ or 2⋅⌈log_{2}(n/3)⌉−1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n−2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(logn) and in Ω(n).

Original language | English |
---|---|

Pages (from-to) | 243-252 |

Number of pages | 10 |

Journal | Computational Geometry: Theory and Applications |

Volume | 68 |

DOIs | |

Publication status | Published - 1 Mar 2018 |

### Fingerprint

### Keywords

- Diameter
- Dual graph
- Optimization
- Simple polygon
- Triangulation

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computational Geometry: Theory and Applications*,

*68*, 243-252. https://doi.org/10.1016/j.comgeo.2017.06.008

**The dual diameter of triangulations.** / Korman, Matias; Langerman, Stefan; Mulzer, Wolfgang; Pilz, Alexander; Saumell, Maria; Vogtenhuber, Birgit.

Research output: Contribution to journal › Article › Research › peer-review

*Computational Geometry: Theory and Applications*, vol. 68, pp. 243-252. https://doi.org/10.1016/j.comgeo.2017.06.008

}

TY - JOUR

T1 - The dual diameter of triangulations

AU - Korman, Matias

AU - Langerman, Stefan

AU - Mulzer, Wolfgang

AU - Pilz, Alexander

AU - Saumell, Maria

AU - Vogtenhuber, Birgit

PY - 2018/3/1

Y1 - 2018/3/1

N2 - Let P be a simple polygon with n vertices. The dual graph T⁎ of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n3logn) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2⋅⌈log2(n/3)⌉ or 2⋅⌈log2(n/3)⌉−1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n−2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(logn) and in Ω(n).

AB - Let P be a simple polygon with n vertices. The dual graph T⁎ of a triangulation T of P is the graph whose vertices correspond to the bounded faces of T and whose edges connect those faces of T that share an edge. We consider triangulations of P that minimize or maximize the diameter of their dual graph. We show that both triangulations can be constructed in O(n3logn) time using dynamic programming. If P is convex, we show that any minimizing triangulation has dual diameter exactly 2⋅⌈log2(n/3)⌉ or 2⋅⌈log2(n/3)⌉−1, depending on n. Trivially, in this case any maximizing triangulation has dual diameter n−2. Furthermore, we investigate the relationship between the dual diameter and the number of ears (triangles with exactly two edges incident to the boundary of P) in a triangulation. For convex P, we show that there is always a triangulation that simultaneously minimizes the dual diameter and maximizes the number of ears. In contrast, we give examples of general simple polygons where every triangulation that maximizes the number of ears has dual diameter that is quadratic in the minimum possible value. We also consider the case of point sets in general position in the plane. We show that for any such set of n points there are triangulations with dual diameter in O(logn) and in Ω(n).

KW - Diameter

KW - Dual graph

KW - Optimization

KW - Simple polygon

KW - Triangulation

UR - http://www.scopus.com/inward/record.url?scp=85023619687&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2017.06.008

DO - 10.1016/j.comgeo.2017.06.008

M3 - Article

VL - 68

SP - 243

EP - 252

JO - Computational geometry

JF - Computational geometry

SN - 0925-7721

ER -