scholarly journals An Upper Bound Asymptotically Tight for the Connectivity of the Disjointness Graph of Segments in the Plane

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1050
Author(s):  
Aurora Espinoza-Valdez ◽  
Jesús Leaños ◽  
Christophe Ndjatchi ◽  
Luis Manuel Ríos-Castro
Keyword(s):  
Upper Bound ◽  
General Position ◽  
Line Segments ◽  
Straight Line ◽  
Point Set

Let P be a set of n≥3 points in general position in the plane. The edge disjointness graph D(P) of P is the graph whose vertices are the n2 closed straight line segments with endpoints in P, two of which are adjacent in D(P) if and only if they are disjoint. In this paper we show that the connectivity of D(P) is at most 7n218+Θ(n), and that this upper bound is asymptotically tight. The proof is based on the analysis of the connectivity of D(Qn), where Qn denotes an n-point set that is almost 3-symmetric.

scholarly journals Edge-Removal and Non-Crossing Configurations in Geometric Graphs

2010 ◽  
Vol Vol. 12 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Oswin Aichholzer ◽  
Sergio Cabello ◽  
Ruy Fabila-Monroy ◽  
David Flores-Peñaloza ◽  
Thomas Hackl ◽  
...  
Keyword(s):  
Extremal Problem ◽  
General Position ◽  
Geometric Graph ◽  
Perfect Matchings ◽  
Geometric Graphs ◽  
Line Segments ◽  
Straight Line ◽  
Edge Removal ◽  
International Audience ◽  
Point Set

Graphs and Algorithms International audience A geometric graph is a graph G = (V, E) drawn in the plane, such that V is a point set in general position and E is a set of straight-line segments whose endpoints belong to V. We study the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum number of removable edges.

PARAMETRIC POLYMATROID OPTIMIZATION AND ITS GEOMETRIC APPLICATIONS

2002 ◽  
Vol 12 (05) ◽  
pp. 429-443 ◽  
Author(s):  
NAOKI KATOH ◽  
HISAO TAMAKI ◽  
TAKESHI TOKUYAMA
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Transportation Problem ◽  
Minimum Weight ◽  
Planar Point ◽  
Line Arrangement ◽  
Optimal Bound ◽  
Straight Line ◽  
Point Set ◽  
Multiple Levels

We give an optimal bound on the number of transitions of the minimum weight base of an integer valued parametric polymatroid. This generalizes and unifies Tamal Dey's O(k1/3 n) upper bound on the number of k-sets (and the complexity of the k-level of a straight-line arrangement), David Eppstein's lower bound on the number of transitions of the minimum weight base of a parametric matroid, and also the Θ(kn) bound on the complexity of the at-most-k level (the union of i-levels for i = 1,2,…,k) of a straight-line arrangement. As applications, we improve Welzl's upper bound on the sum of the complexities of multiple levels, and apply this bound to the number of different equal-sized-bucketings of a planar point set with parallel partition lines. We also consider an application to a special parametric transportation problem.

Circuit presentation and lattice stick number with exactly four z-sticks

2018 ◽  
Vol 27 (08) ◽  
pp. 1850046
Author(s):  
Hyoungjun Kim ◽  
Sungjong No
Keyword(s):  
Upper Bound ◽  
Line Segment ◽  
Disjoint Union ◽  
Lattice Plane ◽  
Cambridge Philos ◽  
Line Segments ◽  
Cubic Lattice ◽  
Straight Line ◽  
Two Component ◽  
Number Formula

The lattice stick number [Formula: see text] of a link [Formula: see text] is defined to be the minimal number of straight line segments required to construct a stick presentation of [Formula: see text] in the cubic lattice. Hong, No and Oh [Upper bound on lattice stick number of knots, Math. Proc. Cambridge Philos. Soc. 155 (2013) 173–179] found a general upper bound [Formula: see text]. A rational link can be represented by a lattice presentation with exactly 4 [Formula: see text]-sticks. An [Formula: see text]-circuit is the disjoint union of [Formula: see text] arcs in the lattice plane [Formula: see text]. An [Formula: see text]-circuit presentation is an embedding obtained from the [Formula: see text]-circuit by connecting each [Formula: see text] pair of vertices with one line segment above the circuit. By using a two-circuit presentation, we can easily find the lattice presentation with exactly four [Formula: see text]-sticks. In this paper, we show that an upper bound for the lattice stick number of rational [Formula: see text]-links realized with exactly four [Formula: see text]-sticks is [Formula: see text]. Furthermore, it is [Formula: see text] if [Formula: see text] is a two-component link.

ALTERNATING HAMILTON CYCLES WITH MINIMUM NUMBER OF CROSSINGS IN THE PLANE

2000 ◽  
Vol 10 (01) ◽  
pp. 73-78 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
M. KANO ◽  
KIYOSHI YOSHIMOTO
Keyword(s):  
Upper Bound ◽  
Time Algorithm ◽  
Hamilton Cycle ◽  
Crossing Number ◽  
Hamilton Cycles ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Disjoint Sets

Let X and Y be two disjoint sets of points in the plane such that |X|=|Y| and no three points of X ∪ Y are on the same line. Then we can draw an alternating Hamilton cycle on X∪Y in the plane which passes through alternately points of X and those of Y, whose edges are straight-line segments, and which contains at most |X|-1 crossings. Our proof gives an O(n2 log n) time algorithm for finding such an alternating Hamilton cycle, where n =|X|. Moreover we show that the above upper bound |X|-1 on crossing number is best possible for some configurations.

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 Upper bound on lattice stick number of knots

2013 ◽  
Vol 155 (1) ◽  
pp. 173-179 ◽  
Author(s):  
KYUNGPYO HONG ◽  
SUNGJONG NO ◽  
SEUNGSANG OH
Keyword(s):  
Upper Bound ◽  
Crossing Number ◽  
Line Segments ◽  
Cubic Lattice ◽  
Straight Line ◽  
Trefoil Knot

AbstractThe lattice stick number sL(K) of a knot K is defined to be the minimal number of straight line segments required to construct a stick presentation of K in the cubic lattice. In this paper, we find an upper bound on the lattice stick number of a nontrivial knot K, except the trefoil knot, in terms of the minimal crossing number c(K) which is sL(K) ≤ 3c(K) + 2. Moreover if K is a non-alternating prime knot, then sL(K) ≤ 3c(K) − 4.

CONSTRAINED POINT-SET EMBEDDABILITY OF PLANAR GRAPHS

2010 ◽  
Vol 20 (05) ◽  
pp. 577-600 ◽  
Author(s):  
EMILIO DI GIACOMO ◽  
WALTER DIDIMO ◽  
GIUSEPPE LIOTTA ◽  
HENK MEIJER ◽  
STEPHEN K. WISMATH
Keyword(s):  
Planar Graph ◽  
General Position ◽  
Planar Graphs ◽  
Disjoint Paths ◽  
Simple Path ◽  
Outerplanar Graph ◽  
Straight Line ◽  
Point Set ◽  
Number Of Bends ◽  
Set Of Points

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

scholarly journals Use of otoliths to estimate size at sexual maturity in fish

2000 ◽  
Vol 43 (4) ◽  
pp. 437-440 ◽  
Author(s):  
Carlos Sérgio Agostinho
Keyword(s):  
Sexual Maturity ◽  
Line Segments ◽  
Alternative Method ◽  
Straight Line ◽  
Size At Sexual Maturity

The viability of an alternative method for estimating the size at sexual maturity of females of Plagioscion squamosissimus (Perciformes, Sciaenidae) was analyzed. This methodology was used to evaluate the size at sexual maturity in crabs, but has not yet been used for this purpose in fishes. Separation of young and adult fishes by this method is accomplished by iterative adjustment of straight-line segments to the data for length of the otolith and length of the fish. The agreement with the estimate previously obtained by another technique and the possibility of calculating the variance indicates that in some cases, the method analyzed can be used successfully to estimate size at sexual maturity in fish. However, additional studies are necessary to detect possible biases in the method.

scholarly journals An Improved Robust Method for Pose Estimation of Cylindrical Parts with Interference Features

Sensors ◽  
2019 ◽  
Vol 19 (10) ◽  
pp. 2234 ◽  
Author(s):  
Jieyu Zhang ◽  
Yuanying Qiu ◽  
Xuechao Duan ◽  
Kangli Xu ◽  
Changqi Yang
Keyword(s):  
Laser Scanning ◽  
Robust Method ◽  
Negative Effects ◽  
Profile Fitting ◽  
Straight Line ◽  
Intelligent Measurement ◽  
Cylindrical Parts ◽  
Point Set ◽  
Aerospace Assembly ◽  
Series Of Experiments

Horizontal docking assembly is a fundamental process in the aerospace assembly, where intelligent measurement and adjustable support systems are urgently needed to achieve higher automation and precision. Thus, a laser scanning approach is employed to obtain the point cloud from a laser scanning sensor. And a method of section profile fitting is put forward to solve the pose parameters from the data cloud acquired by the laser scanning sensor. Firstly, the data is segmented into planar profiles by a series of parallel planes, and ellipse fitting is employed to estimate each center of the section profiles. Secondly, the pose of the part can be obtained through a spatial straight line fitting with these profile centers. However, there may be some interference features on the surface of the parts in the practical assembly process, which will cause negative effects to the measurement. Aiming at the interferences, a robust method improved from M-estimation and RANSAC is proposed to enhance the measurement robustness. The proportion of the inner points in a whole profile point set is set as a judgment criterion to validate each planar profile. Finally, a prototype is fabricated, a series of experiments have been conducted to verify the proposed method.

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