Ramsey-type results for geometric graphs. II

Károlyi, György; Pach, János; Tóth, Géza; Valtr, Pável

doi:10.1145/262839.262908

Cited by 4 publications

(4 citation statements)

References 2 publications

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“…The classical Ramsey number R(n, m) is the smallest integer such that every red-blue complete graph on R(n, m) vertices contains a red K n or a blue K m . The first geometric Ramsey-type problems focused on geometric graphs [19,20] and intersection graphs [24].…”

Section: Previous Work and Resultsmentioning

confidence: 99%

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles

Aichholzer

Huemer

Kappes

et al. 2006

Lecture Notes in Computer Science

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Abstract. We propose a novel subdivision of the plane that consists of both convex polygons and pseudo-triangles. This pseudo-convex decomposition is significantly sparser than either convex decompositions or pseudo-triangulations for planar point sets and simple polygons. We also introduce pseudo-convex partitions and coverings. We establish some basic properties and give combinatorial bounds on their complexity. Our upper bounds depend on new Ramsey-type results concerning disjoint empty convex k-gons in point sets.

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Section: Previous Work and Resultsmentioning

confidence: 99%

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles

Aichholzer

Huemer

Kappes

et al. 2006

Lecture Notes in Computer Science

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“…Every empty convex pentagon or hexagon contains at least three of the four inner points and thus separates the other points, so that no disjoint convex quadrilateral can be found. The coordinates of this point set are: (0, 0), (0, 20), (20,20), (20,0), (1,10), (10,19), (19,10), (10, 1), (5, 7), (7,15), (15,13), (13,5). (19257,42830).…”

Section: Theorem 2 Together With Corollary 1 Impliesmentioning

confidence: 99%

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

Aichholzer

Huemer

Kappes

et al. 2007

Graphs and Combinatorics

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“…The existence of non-crossing Hamiltonian paths and cycles in geometric graphs has been studied in [2,6]. Various Ramsey-type results for non-crossing spanning trees, paths and cycles have been obtained in [14] and [15]. The Euclidean MAX TSP, the problem of computing a longest straight-line tour of a set of points, has been proven NP-hard in dimensions three or higher [12], while its complexity in the Euclidean plane remains open [17].…”

Section: Introductionmentioning

confidence: 99%

Long Non-crossing Configurations in the Plane

Dumitrescu

Tóth

2010

Discrete Comput Geom

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We revisit some maximization problems for geometric networks design under the non-crossing constraint, first studied by Alon, Rajagopalan and Suri (ACM Symposium on Computational Geometry, 1993). Given a set of n points in the plane in general position (no three points collinear), compute a longest non-crossing configuration composed of straight line segments that is: (a) a matching, (b) a Hamiltonian path, and (c) a spanning tree. We obtain some new results for (b) and (c), as well as for the Hamiltonian cycle problem:(i) For the longest non-crossing Hamiltonian path problem, we give an approximation algorithm with ratio 2 π+1 ≈ 0.4829. The previous best ratio, due to Alon et al., was 1 π ≈ 0.3183. The ratio of our algorithm is close to 2 π ≈ 0.6366 on a relatively broad class of instances: for point sets whose perimeter (or diameter) is much shorter than the maximum length matching. For instance "random" point sets meet the condition with high probability. The algorithm runs in O(n 7/3 log n) time.(ii) For the longest non-crossing spanning tree problem, we give an approximation algorithm with ratio 0.502 which runs in O(n log n) time. The previous ratio, 1/2, due to Alon et al., was achieved by a quadratic time algorithm. Along the way, we first re-derive the result of Alon et al. with a faster algorithm and a very simple analysis.(iii) For the longest non-crossing Hamiltonian cycle problem, we give an approximation algorithm whose ratio is close to 2/π on a relatively broad class of instances: for point sets where the product diameter × convex hull size is much smaller than the maximum length matching. Again "random" point sets meet the condition with high probability. However, this algorithm does not come with a constant approximation guarantee for all instances. The algorithm runs in O(n 7/3 log n) time. No previous approximation results were known for this problem.

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Ramsey-type results for geometric graphs. II

Cited by 4 publications

References 2 publications

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-triangles

Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-Triangles

Long Non-crossing Configurations in the Plane

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