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64
The number of different distances determined by n points in the plane
 J. Combin. Theory Ser. A
, 1984
"... A classical problem in combinatorial geometry is that of determining the minimum number f(n) of different distances determined by n points in the Euclidean plane. In 1952, L. Moser proved thatf(n)> n”‘/(2 fi) 1 and this has remained for 30 years as the best lower bound known for f(n). It is shown t ..."
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Cited by 22 (0 self)
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A classical problem in combinatorial geometry is that of determining the minimum number f(n) of different distances determined by n points in the Euclidean plane. In 1952, L. Moser proved thatf(n)> n”‘/(2 fi) 1 and this has remained for 30 years as the best lower bound known for f(n). It is shown that f(n)> cn “ ’ for some fixed constant c. I.
Removing even crossings
 J. COMBINAT. THEORY, SER. B
, 2005
"... An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly ..."
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Cited by 15 (8 self)
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An edge in a drawing of a graph is called even if it intersects every other edge of the graph an even number of times. Pach and Tóth proved that a graph can always be redrawn such that its even edges are not involved in any intersections. We give a new, and significantly simpler, proof of a slightly stronger statement. We show two applications of this strengthened result: an easy proof of a theorem of Hanani and Tutte (the only proof we know of not to use Kuratowski’s theorem), and the result that the odd crossing number of a graph equals the crossing number of the graph for values of at most 3. The paper begins with a disarmingly simple proof of a weak (but standard) version of the theorem by Hanani and Tutte.
The study of translational tiling with Fourier Analysis
, 2003
"... Lectures given at the Workshop on Fourier Analysis and Convexity, Università Di Milano–bicocca ..."
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Cited by 15 (4 self)
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Lectures given at the Workshop on Fourier Analysis and Convexity, Università Di Milano–bicocca
Simultaneous Geometric Graph Embeddings
"... Foundation (JU204/101). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straightline drawing is plana ..."
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Cited by 15 (5 self)
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Foundation (JU204/101). Abstract. We consider the following problem known as simultaneous geometric graph embedding (SGE). Given a set of planar graphs on a shared vertex set, decide whether the vertices can be placed in the plane in such a way that for each graph the straightline drawing is planar. We partially settle an open problem of Erten and Kobourov [5] by showing that even for two graphs the problem is NPhard. We also show that the problem of computing the rectilinear crossing number of a graph can be reduced to a simultaneous geometric graph embedding problem; this implies that placing SGE in NP will be hard, since the corresponding question for rectilinear crossing number is a longstanding open problem. However, rather like rectilinear crossing number, SGE can be decided in PSPACE. 1
Graph Treewidth and Geometric Thickness Parameters
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
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Cited by 14 (8 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and stararboricity.
On the number of plane graphs
 PROC. 17TH ANN. ACMSIAM SYMP. ON DISCRETE ALGORITHMS
, 2006
"... We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extre ..."
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Cited by 8 (1 self)
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We investigate the number of plane geometric, i.e., straightline, graphs, a set S of n points in the plane admits. We show that the number of plane graphs is minimized when S is in convex position, and that the same result holds for several relevant subfamilies. In addition we construct a new extremal configuration, the socalled double zigzag chain. Most noteworthy this example bears Θ ∗ ( √ 72 n) = Θ ∗ (8.4853 n) triangulations and Θ ∗ (41.1889 n) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.
Pushing disks apart – the KneserPoulsen conjecture in the plane, Journal für die Reine und Angewandte Mathematik
, 2002
"... Abstract. We give a proof of the planar case of a longstanding conjecture of Kneser (1955) and Poulsen (1954). In fact, we prove more by showing that if a finite set of disks in the plane is rearranged so that the distance between each pair of centers does not decrease, then the area of the union do ..."
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Cited by 8 (3 self)
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Abstract. We give a proof of the planar case of a longstanding conjecture of Kneser (1955) and Poulsen (1954). In fact, we prove more by showing that if a finite set of disks in the plane is rearranged so that the distance between each pair of centers does not decrease, then the area of the union does not decrease, and the area of the intersection does not increase. 1. Introduction. If p = (p1,...,pN) and q = (q1,...,qN) are two configurations of N points, where each pi ∈ E n and each qi ∈ E n is such that for all 1 ≤ i < j ≤ N, pi − pj  ≤ qi − qj, we say that q is an expansion of p (and p is a contraction of q). If q is an expansion of p, then there may or may not be a continuous motion p(t) = (p1(t),...,pN(t)) for
On Separating Two Simple Polygons by a Single Translation
 Discrete Computational Geometry
, 1989
"... Let P and Q be two disjoint simple polygons having n sides each. We present an algorithm which determines whether Q can be moved by a single translation to a position sufficiently far from P, and which produces all such motions if they exist. The algorithm runs in time O(t(n)) where t(n) is the time ..."
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Cited by 7 (0 self)
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Let P and Q be two disjoint simple polygons having n sides each. We present an algorithm which determines whether Q can be moved by a single translation to a position sufficiently far from P, and which produces all such motions if they exist. The algorithm runs in time O(t(n)) where t(n) is the time needed to triangulate an nsided polygon. Since Tarjan and Van Wyk have recently shown that t(n) = O(n log log n) this improves the previous best result for this problem which was O(n log n) even after triangulation. 1. Introduction Spurred by developments in spatial planning in robotics, computer graphics, and VLSI layout considerable attention has been devoted recently to the problem of moving polygons in the plane without collisions [1][11]. A typical problem in robotics is the FINDPATH problem [12], where a robot must determine if an object, modeled as a polygon in the plane, can be moved from a starting position to a goal state without collisions occurring between the object being m...
An extended lower bound on the number of ( ≤ k)edges to generalized configurations of points and the pseudolinear crossing number of K_n
, 2007
"... ..."
An overview of the Kepler conjecture
"... The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π / √ 18 ≈ 0.74048.... This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th pr ..."
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Cited by 7 (1 self)
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The series of papers in this volume gives a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than π / √ 18 ≈ 0.74048.... This is the oldest problem in discrete geometry and is an important part of Hilbert’s 18th problem. An example of a