scholarly journals More Turán-Type Theorems for Triangles in Convex Point Sets

10.37236/7224 ◽  
2019 ◽  
Vol 26 (1) ◽  
Author(s):  
Boris Aronov ◽  
Vida Dujmović ◽  
Pat Morin ◽  
Aurélien Ooms ◽  
Luı́s Fernando Schultz Xavier da Silveira
Keyword(s):  
Packing Problem ◽  
Point Sets ◽  
Convex Position ◽  
Point Set ◽  
Forbidden Configurations ◽  
Convex Point

 We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations.  As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact.  This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein's longstanding tripod packing problem.

Almost empty polygons

2003 ◽  
Vol 40 (3) ◽  
pp. 269-286 ◽  
Author(s):  
H. Nyklová
Keyword(s):  
Exact Results ◽  
Point Sets ◽  
Planar Point ◽  
Convex Position ◽  
Vertex Set ◽  
Point Set ◽  
Empty Polygons

In this paper we study a problem related to the classical Erdos--Szekeres Theorem on finding points in convex position in planar point sets. We study for which n and k there exists a number h(n,k) such that in every planar point set X of size h(n,k) or larger, no three points on a line, we can find n points forming a vertex set of a convex n-gon with at most k points of X in its interior. Recall that h(n,0) does not exist for n = 7 by a result of Horton. In this paper we prove the following results. First, using Horton's construction with no empty 7-gon we obtain that h(n,k) does not exist for k = 2(n+6)/4-n-3. Then we give some exact results for convex hexagons: every point set containing a convex hexagon contains a convex hexagon with at most seven points inside it, and any such set of at least 19 points contains a convex hexagon with at most five points inside it.

scholarly journals Drawing Hamiltonian Cycles with no Large Angles

10.37236/2356 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Adrian Dumitrescu ◽  
János Pach ◽  
Géza Tóth
Keyword(s):  
Hamiltonian Cycle ◽  
Hamiltonian Cycles ◽  
Point Sets ◽  
Convex Position ◽  
Straight Line ◽  
Open Region ◽  
Point Set ◽  
Rectifiable Curves ◽  
Element Set

Let $n \geq 4$ be even. It is shown that every set $S$ of $n$ points in the plane can be connected by a (possibly self-intersecting) spanning tour (Hamiltonian cycle) consisting of $n$ straight-line edges such that the angle between any two consecutive edges is at most $2\pi/3$. For $n=4$ and $6$, this statement is tight. It is also shown that every even-element point set $S$ can be partitioned  into at most two subsets, $S_1$ and $S_2$, each admitting a spanning tour with no angle larger than $\pi/2$. Fekete and Woeginger conjectured that for sufficiently large even $n$, every $n$-element set admits such a spanning tour. We confirm this conjecture for point sets in convex position. A much stronger result holds for large point sets randomly and uniformly selected from an open region bounded by finitely many rectifiable curves: for any $\epsilon>0$, these sets almost surely admit a spanning tour with no angle larger than $\epsilon$.

scholarly journals Upward Point Set Embeddability for Convex Point Sets Is in P

2012 ◽  
pp. 403-414
Author(s):  
Michael Kaufmann ◽  
Tamara Mchedlidze ◽  
Antonios Symvonis
Keyword(s):  
Point Sets ◽  
Point Set ◽  
Convex Point

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 Fixed poin sets in digital topology, 1

2020 ◽  
Vol 21 (1) ◽  
pp. 87 ◽  
Author(s):  
Laurence Boxer ◽  
P. Christopher Staecker
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Digital Images ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Point Sets ◽  
Fixed Point Set ◽  
Complete Computation ◽  
Point Set ◽  
Fixed Point Sets

<p>In this paper, we examine some properties of the fixed point set of a digitally continuous function. The digital setting requires new methods that are not analogous to those of classical topological fixed point theory, and we obtain results that often differ greatly from standard results in classical topology.</p><p>We introduce several measures related to fixed points for continuous self-maps on digital images, and study their properties. Perhaps the most important of these is the fixed point spectrum F(X) of a digital image: that is, the set of all numbers that can appear as the number of fixed points for some continuous self-map. We give a complete computation of F(C<sub>n</sub>) where C<sub>n</sub> is the digital cycle of n points. For other digital images, we show that, if X has at least 4 points, then F(X) always contains the numbers 0, 1, 2, 3, and the cardinality of X. We give several examples, including C<sub>n</sub>, in which F(X) does not equal {0, 1, . . . , #X}.</p><p>We examine how fixed point sets are affected by rigidity, retraction, deformation retraction, and the formation of wedges and Cartesian products. We also study how fixed point sets in digital images can be arranged; e.g., for some digital images the fixed point set is always connected.</p>

scholarly journals On the length of longest alternating paths for multicoloured point sets in convex position

2006 ◽  
Vol 306 (15) ◽  
pp. 1791-1797 ◽  
Author(s):  
C. Merino ◽  
G. Salazar ◽  
J. Urrutia
Keyword(s):  
Point Sets ◽  
Convex Position

scholarly journals Point Pattern Matching Algorithm for Planar Point Sets under Euclidean Transform

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Xiaoyun Wang ◽  
Xianquan Zhang
Keyword(s):  
Pattern Matching ◽  
Complete Graph ◽  
Point Pattern ◽  
Point Sets ◽  
Matching Problem ◽  
Matching Algorithm ◽  
Planar Point ◽  
Point Pattern Matching ◽  
Point Set ◽  
Pattern Matching Algorithm

Point pattern matching is an important topic of computer vision and pattern recognition. In this paper, we propose a point pattern matching algorithm for two planar point sets under Euclidean transform. We view a point set as a complete graph, establish the relation between the point set and the complete graph, and solve the point pattern matching problem by finding congruent complete graphs. Experiments are conducted to show the effectiveness and robustness of the proposed algorithm.

On a Theorem of Minkowski on Lattice Points in Non-Convex Point Sets

1944 ◽  
Vol 19 (76_Part_4) ◽  
pp. 201-205 ◽  
Author(s):  
K. Mahler
Keyword(s):  
Lattice Points ◽  
Point Sets ◽  
Convex Point

scholarly journals Distribution of Distances and Triangles in a Point Set and Algorithms for Computing the Largest Common Point Sets

1998 ◽  
Vol 20 (3) ◽  
pp. 307-331 ◽  
Author(s):  
T. Akutsu ◽  
H. Tamaki ◽  
T. Tokuyama
Keyword(s):  
Common Point ◽  
Point Sets ◽  
Point Set

scholarly journals INVOLUTIONS OF GRAPH LINK EXTERIORS WHOSE FIXED POINT SETS ARE CLOSED SURFACES

2010 ◽  
Vol 81 (2) ◽  
pp. 298-303 ◽  
Author(s):  
TORU IKEDA
Keyword(s):  
Fixed Point ◽  
Lower Estimate ◽  
Closed Surface ◽  
Point Sets ◽  
Fixed Point Set ◽  
Closed Surfaces ◽  
Point Set ◽  
Graph Link ◽  
Fixed Point Sets

AbstractA link L in S3 possibly admits an involution of the exterior E(L) with fixed point set a closed surface, which is not extendable to an involution of S3. In this paper, we focus on the case of graph links and show that the genus of the surface provides a lower estimate of the number of link components.

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