On the minimum cardinality of a planar point set containing two disjoint convex polygons

2013 ◽  
Vol 50 (3) ◽  
pp. 331-354
Author(s):  
Liping Wu ◽  
Wanbing Lu
Keyword(s):  
Positive Integer ◽  
Convex Hulls ◽  
Convex Polygons ◽  
Minimum Cardinality ◽  
Planar Point ◽  
Point Set ◽  
Disjoint Convex

Let N(k, l) be the smallest positive integer such that any set of N(k, l) points in the plane, no three collinear, contains both a convex k-gon and a convex l-gon with disjoint convex hulls. In this paper, we prove that N(3, 4) = 7, N(4, 4) = 9, N(3, 5) = 10 and N(4, 5) = 11.

scholarly journals On empty convex polygons in a planar point set

2006 ◽  
Vol 113 (3) ◽  
pp. 385-419 ◽  
Author(s):  
Rom Pinchasi ◽  
Radoš Radoičić ◽  
Micha Sharir
Keyword(s):  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Point Set

scholarly journals A Note on the Minimum Size of a Point Set Containing Three Nonintersecting Empty Convex Polygons

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 447
Author(s):  
Qing Yang ◽  
Zengtai You ◽  
Xinshang You
Keyword(s):  
Pairwise Disjoint ◽  
Convex Polygon ◽  
Minimum Size ◽  
Convex Polygons ◽  
Planar Point ◽  
Empty Convex ◽  
Point Set

Let P be a planar point set with no three points collinear, k points of P be a k-hole of P if the k points are the vertices of a convex polygon without points of P. This article proves 13 is the smallest integer such that any planar points set containing at least 13 points with no three points collinear, contains a 3-hole, a 4-hole and a 5-hole which are pairwise disjoint.

ON MAINTAINING THE WIDTH AND DIAMETER OF A PLANAR POINT-SET ONLINE

1993 ◽  
Vol 03 (03) ◽  
pp. 331-344 ◽  
Author(s):  
RAVI JANARDAN
Keyword(s):  
Data Structures ◽  
Online Algorithm ◽  
Greedy Heuristic ◽  
Convex Hulls ◽  
Approximation Formula ◽  
Maximum Weight ◽  
Worst Case ◽  
Planar Point ◽  
Point Set ◽  
The One

Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, S. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α>1 and β≥1 be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(α log 2 n) time, and reports in O(α log 2 n) time an approximation, Ŵ, to the width such that [Formula: see text]. The algorithm for the diameter problem uses O(βn) space, supports updates in O(β log n) time, and reports in O(β) time an approximation, [Formula: see text], to the diameter such that [Formula: see text]. Thus, for instance, even for α as small as 11, Ŵ/W≤1.01, and for β as small as 9, [Formula: see text]. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.

scholarly journals On the number of disjoint convex quadrilaterals for a planar point set

2001 ◽  
Vol 20 (3) ◽  
pp. 97-104 ◽  
Author(s):  
Kiyoshi Hosono ◽  
Masatsugu Urabe
Keyword(s):  
Planar Point ◽  
Point Set ◽  
Disjoint Convex

scholarly journals On the minimum size of a point set containing a 5-hole and a disjoint 4-hole

2011 ◽  
Vol 48 (4) ◽  
pp. 445-457 ◽  
Author(s):  
Bhaswar Bhattacharya ◽  
Sandip Das
Keyword(s):  
Upper Bound ◽  
Minimum Size ◽  
Convex Hulls ◽  
Type Result ◽  
Convex Polygons ◽  
Empty Convex ◽  
Ramsey Type ◽  
Point Set

Let H(k; l), k ≦ l denote the smallest integer such that any set of H(k; l) points in the plane, no three on a line, contains an empty convex k-gon and an empty convex l-gon, which are disjoint, that is, their convex hulls do not intersect. Hosono and Urabe [JCDCG, LNCS 3742, 117–122, 2004] proved that 12 ≦ H(4, 5) ≦ 14. Very recently, using a Ramseytype result for disjoint empty convex polygons proved by Aichholzer et al. [Graphs and Combinatorics, Vol. 23, 481–507, 2007], Hosono and Urabe [Kyoto CGGT, LNCS 4535, 90–100, 2008] improve the upper bound to 13. In this paper, with the help of the same Ramsey-type result, we prove that H(4; 5) = 12.

SEMI-BALANCED PARTITIONS OF TWO SETS OF POINTS AND EMBEDDINGS OF ROOTED FORESTS

2005 ◽  
Vol 15 (03) ◽  
pp. 229-238 ◽  
Author(s):  
ATSUSHI KANEKO ◽  
MIKIO KANO
Keyword(s):  
Positive Integer ◽  
General Position ◽  
Time Algorithm ◽  
Rooted Trees ◽  
Convex Polygons ◽  
Straight Line ◽  
Disjoint Sets ◽  
Rooted Forest ◽  
Balanced Partition ◽  
Disjoint Convex

Let m be a positive integer and let R1, R2 and B be three disjoint sets of points in the plane such that no three points of R1 ∪ R2 ∪ B lie on the same line and |B| = (m-1)|R1|+m|R2|. Put g = |R1∪R2|. Then there exists a subdivision X1∪X2∪⋯∪Xg of the plane into g disjoint convex polygons such that (i) |Xi ∩ (R1 ∪ R2)| = 1 for all 1 ≤ i ≤ g; and (ii) |Xi∩B| = m-1 if |Xi∩R1| = 1, and |Xi∩B| = m if |Xi∩R2| = 1. This partition is called a semi-balanced partition, and our proof gives an O(n4) time algorithm for finding the above semi-balanced partition, where n = |R1| + |R2| + |B|. We next apply the above result to the following theorem: Let T1,…,Tg be g disjoint rooted trees such that |Ti| ∈ {m,m+1} and vi is the root of Ti for all 1 ≤ i ≤ g. Let P be a set of |T1|+⋯+|Tg| points in the plane in general position that contains g specified points p1,…,pg. Then the rooted forest T1 ∪ ⋯ ∪ Tg can be straight-line embedded onto P so that each vi corresponds to pi for every 1 ≤ i ≤ g.

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.

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