Counting Triangulations of Planar Point Sets

Abstract

We study the maximal number of triangulations that a planar set of n points can have, and show that it is at most 30 n . This new bound is achieved by a careful optimization of the charging scheme of , which has led to the previous best upper bound of 43 n for the problem.Moreover, this new bound is useful for bounding the number of other types of planar (i.e., crossing-free) straight-line graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs (O * (239.4 n )), spanning cycles (O * (70.21 n )), spanning trees (160 n ), and cycle-free graphs (O * (202.5 n )).Previous work. Variants of this problem have been studied for over 250 years. The first to consider such a variant was probably Euler, who studied the case of n points in convex position. Euler produced a recursion for the number of triangulations of such sets and guessed its solution, but could not prove its validity. In the 19th century, the problem was studied independently by several mathematicians, which were able to produce some findings, including a proof of Euler's guessed solution. That is, the number of triangulations for the convex case is C n−2 , where C m := 1 m+1 2m m = Θ(m −3/2 4 m ) = Θ * (4 m ), m ∈ N0, is the mth Catalan number 1 (see [22, page 212] for a discussion).During the mid-20th century, Tutte studied several variants of this problem, which did consider points in general position, but had other distinctions from the problem we study (see [23], and [24, pages 114-120]). Avis was perhaps one of the first to ask whether the maximum number of triangulations of n points in the plane is bounded by c n for some c > 0; see [4, page 9]. This fact was established in 1982 by Ajtai, Chvátal, Newborn, and Szemerédi [4], who show that there are at most 10 13n crossing-free graphs on n points-in particular, this bound holds for triangulations.Further developments have yielded progressively better upper bounds for the number of triangulations 2 [21,7,17], so far culminating in the previously mentioned 43 n bound [20] in 2006. This compares to Ω * (8.48 n ), the largest known number of triangulations for a set of n points, derived by Aichholzer et al. [1].The value of tr(n) has also been studied. In a companion paper [18], we derive the bound tr(n) = Ω(2.43 n ) (which improves a previous bound by Aichholzer, Hurtado, and Noy [2]). McCabe and Seidel [11] showed that when the convex hull has only O(1) vertices, there are Ω(2.63 n ) triangulations.Hurtado and Noy [9] presented a configuration of n points in general position and Θ * ( √ 12 n ) ≈

Colored ray configurations

“…A clear motivation for the research in [7] was the extensive investigation on counting noncrossing geometric graphs of several families, such as spanning cycles, perfect matchings, triangulations and many more, and on estimating how large these numbers can get [1,4,9,14,15,16]. On the other hand, arrangements of rays have appeared in graph representation: Ray Intersection Graphs are those in which there is a node for every ray in a given set, two of which are adjacent if they intersect [3,6,17].…”

Counting triangulations and other crossing-free structures approximately

“…Thus any graph traversal algorithm like DFS or BFS can be used to enumerate the vertices of the flip graph of triangulations of P . One limitation of such traversal algorithms, however, is that the amount of memory used is proportional to the number of vertices in the graph -which is known to always be exponential, because the number of triangulations of P lies between Ω(2.43 n ) [29] and O(30 n ) [28]. Using a general technique due to D. Avis and K. Fukuda called Reverse Search, see [10], it is possible to enumerate triangulations while keeping the memory usage polynomial in n. This technique has been further improved in [11,21].…”

Counting and Enumerating Crossing-free Geometric Graphs

“…∀P : O * (c n ) 187.53 [19] 12.24 [22] 10.05 [22] 187.53 [19] 30.00 [18] 141.07 [12] 54.55 [21] ∃P : Ω * (c n ) 41.18 [3] 5.23 [22] 3.00 [11] 8.65 [9] 8.65 [9] 12.52 [13] 4.64 [11] ∀P : Ω * (c n ) 11.65 [10] 3.00 [11] 1 2.00 [11] 2.43 [20] 2.43 [20] 6.75 [10] 1.00 Table 1: Extremal bounds, where cells display the respective exponential bases.…”