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DOI: 10.4230/LIPIcs.SOCG.2015.285
URN: urn:nbn:de:0030-drops-51194
URL: http://drops.dagstuhl.de/opus/volltexte/2015/5119/
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Pilz, Alexander ; Welzl, Emo

Order on Order Types

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Abstract

Given P and P', equally sized planar point sets in general position, we call a bijection from P to P' crossing-preserving if crossings of connecting segments in P are preserved in P' (extra crossings may occur in P'). If such a mapping exists, we say that P' crossing-dominates P, and if such a mapping exists in both directions, P and P' are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossing-dominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomial-time algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.

BibTeX - Entry

@InProceedings{pilz_et_al:LIPIcs:2015:5119,
  author =	{Alexander Pilz and Emo Welzl},
  title =	{{Order on Order Types}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{285--299},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Lars Arge and J{\'a}nos Pach},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2015/5119},
  URN =		{urn:nbn:de:0030-drops-51194},
  doi =		{10.4230/LIPIcs.SOCG.2015.285},
  annote =	{Keywords: point set, order type, planar graph, crossing-free geometric graph}
}

Keywords: point set, order type, planar graph, crossing-free geometric graph
Seminar: 31st International Symposium on Computational Geometry (SoCG 2015)
Issue Date: 2015
Date of publication: 11.06.2015


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