# The Number of Triangulations on Planar Point Sets

```@inproceedings{Welzl2006TheNO,
title={The Number of Triangulations on Planar Point Sets},
author={Emo Welzl},
booktitle={Graph Drawing},
year={2006}
}```
• E. Welzl
• Published in Graph Drawing 18 September 2006
• Mathematics
We give a brief account of results concerning the number of triangulations on finite point sets in the plane, both for arbitrary sets and for specific sets such as the n × n integer lattice.
25 Citations
Counting Polygon Triangulations is Hard
It is proved that it is P-complete to count the triangulations of a (non-simple) polygon.
Happy endings for flip graphs
We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of
On degrees in random triangulations of point sets
• Mathematics
SCG
• 2010
We study the expected number of interior vertices of degree i in a triangulation of a point set S, drawn uniformly at random from the set of all triangulations of S, and derive various bounds and
Chains, Koch Chains, and Point Sets with many Triangulations
• Mathematics
SoCG
• 2022
We introduce the abstract notion of a chain, which is a sequence of n points in the plane, ordered by x -coordinates, so that the edge between any two consecutive points is unavoidable as far as
Minimum number of partial triangulations
• Mathematics
• 2021
We show that the number of partial triangulations of a set of n points on the plane is at least the (n− 2)-nd Catalan number. This is tight for convex n-gons. We also describe all the equality cases.
A bound on a convexity measure for point sets
The main result presented here is a nontrivial combinatorial upper bound of this min-max value in terms of the number of points in the set, and a natural conjecture for the best upper bound is posed.
Counting Triangulations of Planar Point Sets
• Mathematics, Computer Science
Electron. J. Comb.
• 2011
The maximal number of triangulations that a planar set of \$n\$ points can have is shown to be at most \$30^n, which can be used to derive new upper bounds for the number of planar graphs, spanning cycles, spanning trees, and cycle-free graphs.
A Quantitative Steinitz Theorem for Plane Triangulations
• Mathematics
ArXiv
• 2013
It is proved that every plane triangulation \$G\$ with \$n\$ vertices can be embedded in \$\mathbb{R}^2\$ in such a way that it is the vertical projection of a convex polyhedral surface.
Triangulations of nearly convex polygons
• Mathematics, Computer Science
• 2010
The main result shows that the triangulation polynomial, enume rating all triangulations of a nearly convex polygon, is defined in a straightforward way in terms of polynomials associated to the "perturbed" edges.

## References

SHOWING 1-10 OF 27 REFERENCES
Asymptotic Number of Triangulations with Vertices in Z2
Let T2nbe the set of all triangulations of the square [0,n]2 with all the vertices belonging to Z2 with the result that Cn2, the number of vertices in the square, is 2.
Note – On the Number of Triangulations of Planar Point Sets
n points in the plane is at most .
The path of a triangulation
The triangulation path is generalized to (non triangulated) point sets restricted to the interior of simple polygons and proves to be useful for the computation of optimal triangulations.
Encoding a triangulation as a permutation of its point set
• Computer Science
CCCG
• 1997
A new upper bound on the number of triangulations of planar point sets of at most 2 8:2n+O(log n) is obtained.
Studies in computational geometry motivated by mesh generation
This thesis extensively generalizes the famous formula of Heron and Alexandria (75 AD), for the area of a triangle, and presents the first linear time congruence algorithm for 3-dimensional polyhedra.