Order on Order Types

Alexander Pilz, Emo Welzl

Research output: Contribution to journalArticleResearchpeer-review

Abstract

Given P and P, equally sized planar point sets in general position, we call a bijection from P to Pcrossing-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 Pcrossing-dominatesP, 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.

Original languageEnglish
Pages (from-to)886-922
Number of pages37
JournalDiscrete and Computational Geometry
Volume59
Issue number4
DOIs
Publication statusPublished - 1 Jun 2018
Externally publishedYes

Fingerprint

Order Type
Equivalence classes
Point Sets
Polynomials
Partial Order
Bijection
Equivalence class
Convex Hull
Set of points
Polynomial-time Algorithm
Triangular
Sufficient

Keywords

  • Order type
  • Planar graph
  • Point set

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Cite this

Order on Order Types. / Pilz, Alexander; Welzl, Emo.

In: Discrete and Computational Geometry, Vol. 59, No. 4, 01.06.2018, p. 886-922.

Research output: Contribution to journalArticleResearchpeer-review

Pilz, Alexander ; Welzl, Emo. / Order on Order Types. In: Discrete and Computational Geometry. 2018 ; Vol. 59, No. 4. pp. 886-922.
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