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63
The ClarksonShor Technique Revisited and Extended
 Comb., Prob. & Comput
, 2001
"... We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds. ..."
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Cited by 18 (3 self)
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We provide an alternative, simpler and more general derivation of the ClarksonShor probabilistic technique [6] and use it to obtain in addition several extensions and new combinatorial bounds.
Improved Bounds on Planar ksets and klevels
 Discrete Comput. Geom
, 1997
"... We prove an O(nk 1=3 ) upper bound for planar ksets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in arrangement ..."
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Cited by 16 (0 self)
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We prove an O(nk 1=3 ) upper bound for planar ksets. This is the first considerable improvement on this bound after its early solutions approximately twenty seven years ago. Our proof technique also applies to improve the current bounds on the combinatorial complexities of klevels in arrangements of line segments, k convex polygons in the union of n lines, parametric minimum spanning trees and parametric matroids in general. 1 Introduction The problem of determining the optimum asymptotic bound on the number of ksets is one of the most tantalizing open problems in combinatorial geometry. Due to its importance in analyzing geometric algorithms [8, 9, 18], the problem has caught the attention of the computational geometers as well [5, 6, 7, 13, 17, 26, 28]. Given a set P of n points in ! d , and a positive integer k n, a kset is a subset P 0 ` P such that P 0 = P " H for a halfspace H and jP 0 j = k. A close to optimal solution for the problem remains elusive even in ! ...
Crossing Numbers: Bounds and Applications
 I. B'AR'ANY AND K. BOROCZKY, BOLYAI SOCIETY MATHEMATICAL STUDIES 6
, 1997
"... We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the autho ..."
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Cited by 13 (5 self)
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We give a survey of techniques for deriving lower bounds and algorithms for constructing upper bounds for several variations of the crossing number problem. Our aim is to emphasize the more general results or those results which have an algorithmic flavor, including the recent results of the authors. We also show applications of crossing numbers to other areas of discrete mathematics, like discrete geometry.
Crossing Patterns of Segments
, 2001
"... It is shown that for every c > 0 there exists c > 0 satisfying the following condition. Let S be a system of n straightline segments in the plane, which determine at least cn crossings. Then there are two disjoint at least c nelement subsystems, S 1 ; S 2 S, such that every element o ..."
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Cited by 12 (7 self)
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It is shown that for every c > 0 there exists c > 0 satisfying the following condition. Let S be a system of n straightline segments in the plane, which determine at least cn crossings. Then there are two disjoint at least c nelement subsystems, S 1 ; S 2 S, such that every element of S 1 crosses all elements of S 2 .
Geometric Graph Theory
, 1999
"... A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications. ..."
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Cited by 12 (0 self)
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A geometric path is a graph drawn in the plane such that its vertices are points in general position and its edges... This paper surveys some Turántype and Ramseytype extremal problems for geometric graphs, and discusses their generalizations and applications.
Random triangulations of planar points sets
"... Given a set S of n points in the plane, a triangulation is a maximal crossingfree geometric graph on S (in a geometric graph the edges are realized by straight line segments). Here we consider random triangulations, where “random ” refers to uniformly at random from the set of all triangulations of ..."
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Cited by 11 (5 self)
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Given a set S of n points in the plane, a triangulation is a maximal crossingfree geometric graph on S (in a geometric graph the edges are realized by straight line segments). Here we consider random triangulations, where “random ” refers to uniformly at random from the set of all triangulations of S. We are primarily interested in the degree sequences of such random triangulations.
Crossing number, paircrossing number, and expansion
 J. Combin. Theory Ser. B
, 2004
"... We also prove by similar methods that a graph G with crossing number k = cr(G) ? Cpssqd(G) m log2 n has a nonplanar subgraph on at most O\Gamma \Delta nm log2 nk \Delta vertices, where m is the number of edges, \Delta is the maximum degree in G, and C is a suitable sufficiently large constant. ..."
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Cited by 10 (0 self)
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We also prove by similar methods that a graph G with crossing number k = cr(G) ? Cpssqd(G) m log2 n has a nonplanar subgraph on at most O\Gamma \Delta nm log2 nk \Delta vertices, where m is the number of edges, \Delta is the maximum degree in G, and C is a suitable sufficiently large constant.
Counting Triangulations of Planar Point Sets
"... 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. More ..."
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Cited by 9 (3 self)
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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., crossingfree) straightline graphs on a given point set. Specifically, we derive new upper bounds for the number of planar graphs (O ∗ (239.4 n)), spanning cycles (O ∗ (70.21 n)), spanning trees (160 n), and cyclefree graphs (O ∗ (202.5 n)).