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66
Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
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Cited by 113 (1 self)
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We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
Arrangements and Their Applications
 Handbook of Computational Geometry
, 1998
"... The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arr ..."
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Cited by 78 (22 self)
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The arrangement of a finite collection of geometric objects is the decomposition of the space into connected cells induced by them. We survey combinatorial and algorithmic properties of arrangements of arcs in the plane and of surface patches in higher dimensions. We present many applications of arrangements to problems in motion planning, visualization, range searching, molecular modeling, and geometric optimization. Some results involving planar arrangements of arcs have been presented in a companion chapter in this book, and are extended in this chapter to higher dimensions. Work by P.A. was supported by Army Research Office MURI grant DAAH049610013, by a Sloan fellowship, by an NYI award, and by a grant from the U.S.Israeli Binational Science Foundation. Work by M.S. was supported by NSF Grants CCR9122103 and CCR9311127, by a MaxPlanck Research Award, and by grants from the U.S.Israeli Binational Science Foundation, the Israel Science Fund administered by the Israeli Ac...
Planarizing Graphs  A Survey and Annotated Bibliography
, 1999
"... Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results abo ..."
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Cited by 33 (0 self)
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Given a finite, undirected, simple graph G, we are concerned with operations on G that transform it into a planar graph. We give a survey of results about such operations and related graph parameters. While there are many algorithmic results about planarization through edge deletion, the results about vertex splitting, thickness, and crossing number are mostly of a structural nature. We also include a brief section on vertex deletion. We do not consider parallel algorithms, nor do we deal with online algorithms.
On the Number of Incidences Between Points and Curves
 Combinatorics, Probability and Computing 7
"... We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane. ..."
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Cited by 30 (14 self)
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We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane.
A Quadratic Time Algorithm for the MinMax Length Triangulation
 SIAM J. Comput
, 1991
"... Abstract. We show that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n 2). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative n ..."
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Cited by 28 (3 self)
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Abstract. We show that a triangulation of a set of n points in the plane that minimizes the maximum edge length can be computed in time O(n 2). The algorithm is reasonably easy to implement and is based on the theorem that there is a triangulation with minmax edge length that contains the relative neighborhood graph of the points as a subgraph. With minor modi cations the algorithm works for arbitrary normed metrics. Key words. Computational geometry,point sets, triangulations, two dimensions, minmax edge length, normed metrics AMS(MOS) subject classi cations. 68U05, 68Q25, 65D05 Appear in: SIAM Journal on Computing, 22 (3), 527{551, (1993)
A better upper bound on the number of triangulations of a planar point set
 Journal of Combinatorial Theory, Ser. A
"... Abstract. We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straightedge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)). ..."
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Cited by 25 (3 self)
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Abstract. We show that a point set of cardinality n in the plane cannot be the vertex set of more than 59 n O(n −6) straightedge triangulations of its convex hull. This improves the previous upper bound of 276.75 n+O(log(n)).
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
, 2006
"... Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and com ..."
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Cited by 25 (2 self)
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Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c> 1024/31827> 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 25 e − (v − 2). Both bounds are tight upt o
Extremal Problems for Geometric Hypergraphs
 Discrete Comput. Geom
, 1998
"... A geometric hypergraph H is a collection of idimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)tuples of a vertex set V in general position in dspace. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it ..."
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Cited by 24 (2 self)
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A geometric hypergraph H is a collection of idimensional simplices, called hyperedges or, simply, edges, induced by some (i + 1)tuples of a vertex set V in general position in dspace. The topological structure of geometric graphs, i.e., the case d = 2; i = 1, has been studied extensively, and it proved to be instrumental for the solution of a wide range of problems in combinatorial and computational geometry. They include the kset problem, proximity questions, bounding the number of incidences between points and lines, designing various efficient graph drawing algorithms, etc. In this paper, we make an attempt to generalize some of these tools to higher dimensions. We will mainly consider extremal problems of the following type. What is the largest number of edges (isimplices) that a geometric hypergraph of n vertices can have without containing certain forbidden configurations? In particular, we discuss the special cases when the forbidden configurations are k intersecting edges...
Cutting Circles into Pseudosegments and Improved Bounds for Incidences
 Geom
, 2000
"... We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m poi ..."
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Cited by 23 (12 self)
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We show that n arbitrary circles in the plane can be cut into O(n 3/2+# ) arcs, for any # > 0, such that any pair of arcs intersect at most once. This improves a recent result of Tamaki and Tokuyama [20]. We use this result to obtain improved upper bounds on the number of incidences between m points and n circles. An improved incidence bound is also obtained for graphs of polynomials of any constant maximum degree. 1 Introduction Let P be a finite set of points in the plane and C a finite set of circles. Let I = I(P, C) denote the number of incidences between the points and the circles. Let I(m, n) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles, and let I # (m, n, X) denote the maximum value of I(P, C), taken over all sets P of m points and sets C of n circles with at most X intersecting pairs. In this paper we derive improved upper bounds for I(m, n) and I # (m, n, X). The previous best upper bounds were I(m, n) = O(m 3/5 n 4/5 ...
On the Number of CrossingFree Matchings, Cycles, and Partitions
, 2006
"... We show that a set of n points in the plane has at most O(10.05n) perfect matchings with crossingfree straightline embedding. The expected number of perfect crossingfree matchings of a set of n points drawn i.i.d. from an arbitrary distribution in the plane is at most ..."
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Cited by 19 (3 self)
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We show that a set of n points in the plane has at most O(10.05n) perfect matchings with crossingfree straightline embedding. The expected number of perfect crossingfree matchings of a set of n points drawn i.i.d. from an arbitrary distribution in the plane is at most