Packing Plane Perfect Matchings into a Point Set

International audience Given a set $P$ of $n$ points in the plane, where $n$ is even, we consider the following question: How many plane perfect matchings can be packed into $P$? For points in general position we prove the lower bound of ⌊log2$n$⌋$-1$. For some special configurations of point sets, we give the exact answer. We also consider some restricted variants of this problem.

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Edge-Removal and Non-Crossing Configurations in Geometric Graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.525 ◽

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.

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On the numbers of radial orderings of planar point sets

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2087 ◽

2014 ◽

Vol Vol. 16 no. 3 (Combinatorics) ◽

Author(s):

José Miguel Dıaz-Banez ◽

Ruy Fabila-Monroy ◽

Pablo Pérez-Lantero

Keyword(s):

General Position ◽

Point Sets ◽

Planar Point ◽

International Audience

Combinatorics International audience Given a set S of n points in the plane, a radial ordering of S with respect to a point p (not in S) is a clockwise circular ordering of the elements in S by angle around p. If S is two-colored, a colored radial ordering is a radial ordering of S in which only the colors of the points are considered. In this paper, we obtain bounds on the number of distinct non-colored and colored radial orderings of S. We assume a strong general position on S, not three points are collinear and not three lines 14;each passing through a pair of points in S 14;intersect in a point of ℝ2 S. In the colored case, S is a set of 2n points partitioned into n red and n blue points, and n is even. We prove that: the number of distinct radial orderings of S is at most O(n4) and at least Ω(n3); the number of colored radial orderings of S is at most O(n4) and at least Ω(n); there exist sets of points with Θ(n4) colored radial orderings and sets of points with only O(n2) colored radial orderings.

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NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195914600012 ◽

2014 ◽

Vol 24 (03) ◽

pp. 177-181 ◽

Cited By ~ 1

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.

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Almost empty polygons

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.40.2003.3.1 ◽

2003 ◽

Vol 40 (3) ◽

pp. 269-286 ◽

Cited By ~ 4

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.

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An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽

10.3390/math9050525 ◽

2021 ◽

Vol 9 (5) ◽

pp. 525

Author(s):

Javier Rodrigo ◽

Susana Merchán ◽

Danilo Magistrali ◽

Mariló López

Keyword(s):

Lower Bound ◽

Complete Graph ◽

General Position ◽

Crossing Number ◽

Minimum Number

In this paper, we improve the lower bound on the minimum number of ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

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Finding Points in General Position

International Journal of Computational Geometry & Applications ◽

10.1142/s021819591750008x ◽

2017 ◽

Vol 27 (04) ◽

pp. 277-296 ◽

Cited By ~ 6

Author(s):

Vincent Froese ◽

Iyad Kanj ◽

André Nichterlein ◽

Rolf Niedermeier

Keyword(s):

Lower Bound ◽

General Position ◽

Subset Selection ◽

Selection Problem ◽

Fixed Parameter Tractability ◽

Maximum Cardinality ◽

Exponential Time ◽

Running Time ◽

Fixed Parameter ◽

Set Of Points

We study the General Position Subset Selection problem: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and present several fixed-parameter tractability results for the problem as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.

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Fixed poin sets in digital topology, 1

Applied General Topology ◽

10.4995/agt.2020.12091 ◽

2020 ◽

Vol 21 (1) ◽

pp. 87 ◽

Cited By ~ 3

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

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.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(Cn) where Cn 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 Cn, in which F(X) does not equal {0, 1, . . . , #X}.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.

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On Greedy Trie Execution

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3007 ◽

2012 ◽

Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽

Author(s):

Zbigniew Gołębiewski ◽

Filip Zagórski

Keyword(s):

Lower Bound ◽

Greedy Algorithm ◽

International Audience ◽

The Greedy Algorithm

International audience In the paper "How to select a looser'' Prodinger was analyzing an algorithm where $n$ participants are selecting a leader by flipping <underline>fair</underline> coins, where recursively, the 0-party (those who i.e. have tossed heads) continues until the leader is chosen. We give an answer to the question stated in the Prodinger's paper – what happens if not a 0-party is recursively looking for a leader but always a party with a smaller cardinality. We show the lower bound on the number of rounds of the greedy algorithm (for <underline>fair</underline> coin).

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PARAMETRIC POLYMATROID OPTIMIZATION AND ITS GEOMETRIC APPLICATIONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195902000967 ◽

2002 ◽

Vol 12 (05) ◽

pp. 429-443 ◽

Cited By ~ 3

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.

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Planar point sets with a small number of empty convex polygons

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.41.2004.2.4 ◽

2004 ◽

Vol 41 (2) ◽

pp. 243-269 ◽

Cited By ~ 5

Author(s):

Imre Bárány ◽

Pável Valtr

Keyword(s):

Convex Hull ◽

General Position ◽

Point Sets ◽

Convex Polygons ◽

Planar Point ◽

Empty Convex ◽

Finite Set ◽

Empty Triangles

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n2 empty triangles, ˜1.94n2 empty quadrilaterals, ˜1.02n2 empty pentagons, and ˜0.2n2 empty hexagons.

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