## A better upper bound on the number of triangulations of a planar point set

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Venue: | Journal of Combinatorial Theory, Ser. A |

Citations: | 25 - 3 self |

### BibTeX

@ARTICLE{Santos_abetter,

author = {Francisco Santos and Raimund Seidel},

title = {A better upper bound on the number of triangulations of a planar point set},

journal = {Journal of Combinatorial Theory, Ser. A},

year = {}

}

### Years of Citing Articles

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### Abstract

Abstract. We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straight-edge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)).

### Citations

85 |
Constructions and complexity of secondary polytopes
- Billera, Filliman, et al.
- 1990
(Show Context)
Citation Context ... ( v+b+5 6 ) � v+b+5 6 � v+b+6 6 (6 + v + b)N ≤ (v + b) 59v · 7b � = (6 + v + b) 59v · 7b �. . Hence, Remark 5. In the theory of secondary polytopes and the so-called Baues problem (see, for example, =-=[4]-=- and [8]) it is natural to consider as triangulations of A those covering conv(A) and with vertex set contained in A, allowing not to use all of the interior points as vertices. We can bound the numbe... |

71 |
Crossing-free subgraphs
- Ajtai, Chvátal, et al.
- 1982
(Show Context)
Citation Context .... In this paper we prove that a point set of cardinality n cannot have more than 59 n O(n −6 ) triangulations. An upper bound of type 2 O(n) for this number is a consequence of the general results of =-=[3]-=-. Upper bounds of 173 000 n , 7 187.52 n and 276.75 n+O(log(n)) have been given, respectively, in [10], [9] and [5]. The precise statement of our new upper bound is: Theorem 1. The number of triangula... |

47 | Encoding a triangulation as a permutation of its point set
- Denny, Sohler
- 1997
(Show Context)
Citation Context ...pper bound of type 2 O(n) for this number is a consequence of the general results of [3]. Upper bounds of 173 000 n , 7 187.52 n and 276.75 n+O(log(n)) have been given, respectively, in [10], [9] and =-=[5]-=-. The precise statement of our new upper bound is: Theorem 1. The number of triangulations of a planar point set is bounded above by 59 v · 7 b � v+b+6 6 where v and b denote the numbers of interior a... |

45 |
Studies in Computational Geometry Motivated by Mesh Generation
- Smith
- 1989
(Show Context)
Citation Context ...ulations. An upper bound of type 2 O(n) for this number is a consequence of the general results of [3]. Upper bounds of 173 000 n , 7 187.52 n and 276.75 n+O(log(n)) have been given, respectively, in =-=[10]-=-, [9] and [5]. The precise statement of our new upper bound is: Theorem 1. The number of triangulations of a planar point set is bounded above by 59 v · 7 b � v+b+6 6 where v and b denote the numbers ... |

27 |
Lower bounds on the number of crossing-free subgraphs of Kn
- García, Noy, et al.
- 2000
(Show Context)
Citation Context ... number of triangulations known. The number of triangulations of each, for n = 18, is shown. � triangulations � each, and a non-convex 2k-gon which is easily seen to have 2k−2 k−1 triangulations (see =-=[6]-=-). Hence, the number of triangulations of A is: � � 2k − 2 Ck−2 k − 1 2 = Θ(64 k 7 − k 2) = Θ(8 n n −7 2 ). • A double circle: Let A be a convex k-gon (k = n/2) together with an interior point suffici... |

17 | The generalized Baues problem - Reiner - 1999 |

12 |
Counting triangulations of almost-convex polygons, Ars Combin. 45
- Hurtado, Noy
- 1997
(Show Context)
Citation Context ...ure 1, for the case k = 9. Again, the edges drawn are unavoidable, and triangulating A is the same as triangulating the central non-convex n-gon. An inclusion-exclusion argument (see Proposition 1 in =-=[7]-=-) gives the exact number of triangulations of this polygon, which is k� (−1) i � � k C2k−i−2 ≤ 12 i k = 12 n 2 . i=0 It is interesting to observe that the double circle actually gives the minimum poss... |

12 |
The generalized Baues problem. New perspectives in algebraic combinatorics
- Reiner
(Show Context)
Citation Context ... 6 ) � v+b+5 6 � v+b+6 6 (6 + v + b)N ≤ (v + b) 59v · 7b � = (6 + v + b) 59v · 7b �. . Hence, Remark 5. In the theory of secondary polytopes and the so-called Baues problem (see, for example, [4] and =-=[8]-=-) it is natural to consider as triangulations of A those covering conv(A) and with vertex set contained in A, allowing not to use all of the interior points as vertices. We can bound the number of tri... |

8 | On the number of triangulations every planar point set must have
- Aichholzer, Hurtado, et al.
- 2001
(Show Context)
Citation Context ...wn about the maximum and minimum number of triangulations of point sets of fixed cardinality. In particular, we mention that every point set in general position has at least Ω(2.012 n ), as proved in =-=[2]-=-. As a reference, compare these upper and lower bounds to the number of triangulations � of n points in convex position, which is the n − = Θ(4 n 3 2). Catalan number Cn−2 = 1 �2n−4 n−1 n−2 �, 1. Proo... |

8 | On the number of triangulations of planar point sets, Combinatorica 18 - Seidel - 1998 |

6 |
On the number of triangulations of planar point sets
- Seidel
- 1998
(Show Context)
Citation Context ...ns. An upper bound of type 2 O(n) for this number is a consequence of the general results of [3]. Upper bounds of 173 000 n , 7 187.52 n and 276.75 n+O(log(n)) have been given, respectively, in [10], =-=[9]-=- and [5]. The precise statement of our new upper bound is: Theorem 1. The number of triangulations of a planar point set is bounded above by 59 v · 7 b � v+b+6 6 where v and b denote the numbers of in... |

4 |
The point-set order-type database: a collection of applications and results
- Aicholzer, Krasser
- 2001
(Show Context)
Citation Context ...s necessary to assume general position since n − 1 points on a line produce a point set with only 1 triangulation. For T(n), general position is no loss of generality. The following table, taken from =-=[1]-=-, gives T(n) and t(n) for n = 3, . . .,10, compared to the number of triangulations of the convex n-gon: (1) (2) n 3 4 5 6 7 8 9 10 t(n) 1 1 2 4 11 30 89 250 Cn−2 1 2 5 14 42 132 429 1430 T(n) 1 2 5 1... |