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Abstract order type extension and new results on the rectilinear crossing number

Published: 06 June 2005 Publication History

Abstract

We extend the order type data base of all realizable order types in the plane to point sets of cardinality 11. More precisely, we provide a complete data base of all combinatorial different sets of up to 11 points in general position in the plane. In addition, we develop a novel and efficient method for a complete extension to order types of size 12 and more in an abstract sense, that is, without the need to store or realize the sets. The presented method is well suited for independent computations. Thus, time intensive investigations benefit from the possibility of distributed computing.Our approach has various applications to combinatorial problems which are based on sets of points in the plane. This includes classic problems like searching for (empty) convex k-gons ('happy end problem'), decomposing sets into convex regions, counting structures like triangulations or pseudo-triangulations, minimal crossing numbers, and more. We present some improved results to all these problems. As an outstanding result we have been able to determine the exact rectilinear crossing number of the complete graph Kn for up to n = 17, the largest previous range being n = 12, and slightly improved the asymptotic upper bound.

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cover image ACM Conferences
SCG '05: Proceedings of the twenty-first annual symposium on Computational geometry
June 2005
398 pages
ISBN:1581139918
DOI:10.1145/1064092
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 06 June 2005

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Author Tags

  1. complete graph
  2. order type
  3. pseudoline arrangement
  4. rectilinear crossing number

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SoCG05

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SCG '05 Paper Acceptance Rate 41 of 141 submissions, 29%;
Overall Acceptance Rate 625 of 1,685 submissions, 37%

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  • (2014)The complexity of order type isomorphismProceedings of the twenty-fifth annual ACM-SIAM symposium on Discrete algorithms10.5555/2634074.2634104(405-415)Online publication date: 5-Jan-2014
  • (2007)Abstract order type extension and new results on the rectilinear crossing numberComputational Geometry: Theory and Applications10.1016/j.comgeo.2005.07.00536:1(2-15)Online publication date: 1-Jan-2007
  • (2007)Decompositions, Partitions, and Coverings with Convex Polygons and Pseudo-TrianglesGraphs and Combinatorics10.1007/s00373-007-0752-x23:5(481-507)Online publication date: 1-Oct-2007
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  • (2006)On the Crossing Number of Complete GraphsComputing10.1007/s00607-005-0133-376:1(165-176)Online publication date: 1-Jan-2006
  • (2006)Decompositions, partitions, and coverings with convex polygons and pseudo-trianglesProceedings of the 31st international conference on Mathematical Foundations of Computer Science10.1007/11821069_8(86-97)Online publication date: 28-Aug-2006

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