11 research outputs found
A better upper bound on the number of triangulations of a planar point set
We show that a point set of cardinality in the plane cannot be the vertex
set of more than straight-edge triangulations of its convex
hull. This improves the previous upper bound of .Comment: 6 pages, 1 figur
Counting Triangulations and other Crossing-Free Structures Approximately
We consider the problem of counting straight-edge triangulations of a given
set of points in the plane. Until very recently it was not known
whether the exact number of triangulations of can be computed
asymptotically faster than by enumerating all triangulations. We now know that
the number of triangulations of can be computed in time,
which is less than the lower bound of on the number of
triangulations of any point set. In this paper we address the question of
whether one can approximately count triangulations in sub-exponential time. We
present an algorithm with sub-exponential running time and sub-exponential
approximation ratio, that is, denoting by the output of our
algorithm, and by the exact number of triangulations of , for some
positive constant , we prove that . This is the first algorithm that in sub-exponential time computes a
-approximation of the base of the number of triangulations, more
precisely, . Our algorithm can be
adapted to approximately count other crossing-free structures on , keeping
the quality of approximation and running time intact. In this paper we show how
to do this for matchings and spanning trees.Comment: 19 pages, 2 figures. A preliminary version appeared at CCCG 201
On the Number of Pseudo-Triangulations of Certain Point Sets
We pose a monotonicity conjecture on the number of pseudo-triangulations of
any planar point set, and check it on two prominent families of point sets,
namely the so-called double circle and double chain. The latter has
asymptotically pointed pseudo-triangulations, which lies
significantly above the maximum number of triangulations in a planar point set
known so far.Comment: 31 pages, 11 figures, 4 tables. Not much technical changes with
respect to v1, except some proofs and statements are slightly more precise
and some expositions more clear. This version has been accepted in J. Combin.
Th. A. The increase in number of pages from v1 is mostly due to formatting
the paper with "elsart.cls" for Elsevie
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
A new lower bound on the maximum number of plane graphs using production matrices
© 2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We use the concept of production matrices to show that there exist sets of n points in the plane that admit ¿(42.11n ) crossing-free geometric graphs. This improves the previously best known bound of ¿(41.18n ) by Aichholzer et al. (2007).Postprint (author's final draft
On the number of drawings of a combinatorial triangulations
Aquesta tesi explora la relació entre triangulacions combinatòries i geomètriques en geometria discreta i combinatòria. Per triangulacions combinatòries ens referim a grafs, mentre que per triangulacions geomètriques ens referim a dibuixos de grafs com a plans maximals amb línies rectes com a arestes sobre un conjunt de punts fixat al pla. Estudiem de quantes maneres es pot traçar una triangulació combinatòria com a triangulació geomètrica sobre un conjunt de punts donat. La nostra contribució central és demostrar que una triangulació combinatòria fixa amb n vèrtexs es pot dibuixar d'almenys 1,31^n maneres en un conjunt de n punts com a diferents triangulacions geomètriques. També analitzem els límits superiors i una versió acolorida daquest problema. L'enfocament suggerit pot ajudar a avançar en la resolució del problema obert per limitar el nombre de triangulacions geomètriques.
A més, aprofundim en fonaments històrics, com el treball de Tutte, que proporciona el nombre exacte de triangulacions combinatòries amb n vèrtexs.Esta tesis explora la relación entre triangulaciones combinatorias y geométricas en geometría discreta y combinatoria. Con triangulaciones combinatorias nos referimos a grafos, mientras que con triangulaciones geométricas nos referimos a dibujos de grafos como planos maximales con líneas rectas como aristas sobre un conjunto de puntos fijado en el plano. Estudiamos de cuántas maneras se puede trazar una triangulación combinatoria como triangulación geométrica sobre un conjunto de puntos dado. Nuestra contribución central es demostrar que una triangulación combinatoria fija con n vértices se puede dibujar de al menos 1,31^n maneras en un conjunto de n puntos como diferentes triangulaciones geométricas. También analizamos los límites superiores y una versión coloreada de este problema. El enfoque sugerido puede ayudar a avanzar en la resolución del problema abierto para limitar el número de triangulaciones geométricas.
Además, profundizamos en fundamentos históricos, como el trabajo de Tutte, que proporciona el número exacto de triangulaciones combinatorias con n vértices.This thesis explores the intricate relationship between combinatorial and geometric triangulations in discrete and combinatorial geometry. With combinatorial triangulations we refer to graphs, while with geometric triangulations we refer to maximal planar straight-line drawings on a point set in the plane. We study in how many ways can a combinatorial triangulation be drawn as geometric triangulation on a given point set. Our central contribution is proving that a fixed combinatorial triangulation with n vertices can be drawn in at least 1.31^n ways in a set of n points as different geometric triangulations. We also discuss upper bounds and a colored version of this problem. The suggested approach may help to advance the resolution of the open problem to bound the number of geometric triangulations.
Additionally, we delve into historical foundations, such as Tutte's work, which provides the exact number of combinatorial triangulations with n vertices