SIMULTANEOUS EDGE FLIPPING IN TRIANGULATIONS

2003 ◽  
Vol 13 (02) ◽  
pp. 113-133 ◽  
Author(s):  
JERÔME GALTIER ◽  
FERRAN HURTADO ◽  
MARC NOY ◽  
STÉPHANE PÉRENNES ◽  
JORGE URRUTIA
Keyword(s):  
Upper Bound ◽  
Point Sets

We generalize the operation of flipping an edge in a triangulation to that of flipping several edges simultaneously. Our main result is an optimal upper bound on the number of simultaneous flips that are needed to transform a triangulation into another. Our results hold for triangulations of point sets and for polygons.

scholarly journals Counting Triangulations of Planar Point Sets

10.37236/557 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Micha Sharir ◽  
Adam Sheffer
Keyword(s):  
Upper Bound ◽  
Planar Graphs ◽  
Spanning Trees ◽  
Upper Bounds ◽  
Line Graphs ◽  
Point Sets ◽  
Planar Point ◽  
Straight Line ◽  
Point Set ◽  
Spanning Cycles

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 Sharir and Welzl (2006), 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, it can be used to derive new upper bounds for the number of planar graphs ($207.84^n$), spanning cycles ($O(68.67^n)$), spanning trees ($O(146.69^n)$), and cycle-free graphs ($O(164.17^n)$).

scholarly journals A Tight Lower Bound for Convexly Independent Subsets of the Minkowski Sums of Planar Point Sets

10.37236/484 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Ondřej Bílka ◽  
Kevin Buchin ◽  
Radoslav Fulek ◽  
Masashi Kiyomi ◽  
Yoshio Okamoto ◽  
...  
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Minkowski Sum ◽  
Independent Subset ◽  
Point Sets ◽  
Planar Point ◽  
Minkowski Sums

Recently, Eisenbrand, Pach, Rothvoß, and Sopher studied the function $M(m, n)$, which is the largest cardinality of a convexly independent subset of the Minkowski sum of some planar point sets $P$ and $Q$ with $|P| = m$ and $|Q| = n$. They proved that $M(m,n)=O(m^{2/3}n^{2/3}+m+n)$, and asked whether a superlinear lower bound exists for $M(n,n)$. In this note, we show that their upper bound is the best possible apart from constant factors.

scholarly journals ON POINT SETS IN VECTOR SPACES OVER FINITE FIELDS THAT DETERMINE ONLY ACUTE ANGLE TRIANGLES

2009 ◽  
Vol 81 (1) ◽  
pp. 114-120 ◽  
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Upper Bound ◽  
Dimensional Space ◽  
Real Space ◽  
Acute Angle ◽  
Vector Spaces ◽  
Point Sets ◽  
Similar Question ◽  
The Real

AbstractFor three points$\vec {u}$,$\vec {v}$and$\vec {w}$in then-dimensional space 𝔽nqover the finite field 𝔽qofqelements we give a natural interpretation of an acute angle triangle defined by these points. We obtain an upper bound on the size of a set 𝒵 such that all triples of distinct points$\vec {u}, \vec {v}, \vec {w} \in \cZ $define acute angle triangles. A similar question in the real space ℛndates back to P. Erdős and has been studied by several authors.

scholarly journals A Bound on a Convexity Measure for Point Sets

2019 ◽  
Vol 29 (04) ◽  
pp. 301-306
Author(s):  
Danny Rorabaugh
Keyword(s):  
Upper Bound ◽  
Interior Angle ◽  
Point Sets ◽  
Planar Point ◽  
Angle Measure ◽  
Convex Position ◽  
Point Set ◽  
Set Of Points ◽  
Convexity Measure

A planar point set is in convex position precisely when it has a convex polygonization, that is, a polygonization with maximum interior angle measure at most [Formula: see text]. We can thus talk about the convexity of a set of points in terms of its min-max interior angle measure. The main result presented here is a nontrivial upper bound of the min-max value in terms of the number of points in the set. Motivated by a particular construction, we also pose a natural conjecture for the best upper bound.

scholarly journals An upper bound for the minimum diameter of integral point sets

1993 ◽  
Vol 9 (4) ◽  
pp. 427-432 ◽  
Author(s):  
Heiko Harborth ◽  
Arnfried Kemnitz ◽  
Meinhard Möller
Keyword(s):  
Upper Bound ◽  
Integral Point ◽  
Point Sets ◽  
Minimum Diameter

scholarly journals NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

2014 ◽  
Vol 24 (03) ◽  
pp. 177-181 ◽  
Author(s):  
RUY FABILA-MONROY ◽  
CLEMENS HUEMER ◽  
EULÀLIA TRAMUNS
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
General Position ◽  
Crossing Number ◽  
The Other ◽  
Point Sets ◽  
Convex Position ◽  
Minimum Number

In 1979 Conway, Croft, Erdős and Guy proved that every set S of n points in general position in the plane determines at least [Formula: see text] obtuse angles and also presented a special set of n points to show the upper bound [Formula: see text] on the minimum number of obtuse angles among all sets S. We prove that every set S of n points in convex position determines at least [Formula: see text] obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.

scholarly journals On convex holes in d-dimensional point sets

2021 ◽  
pp. 1-8
Author(s):  
Boris Bukh ◽  
Ting-Wei Chao ◽  
Ron Holzman
Keyword(s):  
Numerical Analysis ◽  
Upper Bound ◽  
General Position ◽  
Convex Polytope ◽  
Point Sets ◽  
Finite Set ◽  
Basic Version

Abstract Given a finite set $A \subseteq \mathbb{R}^d$ , points $a_1,a_2,\dotsc,a_{\ell} \in A$ form an $\ell$ -hole in A if they are the vertices of a convex polytope, which contains no points of A in its interior. We construct arbitrarily large point sets in general position in $\mathbb{R}^d$ having no holes of size $O(4^dd\log d)$ or more. This improves the previously known upper bound of order $d^{d+o(d)}$ due to Valtr. The basic version of our construction uses a certain type of equidistributed point sets, originating from numerical analysis, known as (t,m,s)-nets or (t,s)-sequences, yielding a bound of $2^{7d}$ . The better bound is obtained using a variant of (t,m,s)-nets, obeying a relaxed equidistribution condition.

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