Order on Order Types

Alexander Pilz*, Emo Welzl

*Korrespondierende/r Autor/in für diese Arbeit

Publikation: Beitrag in einer FachzeitschriftArtikel


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.

Seiten (von - bis)886-922
FachzeitschriftDiscrete & Computational Geometry
PublikationsstatusVeröffentlicht - 1 Jun 2018
Extern publiziertJa

ASJC Scopus subject areas

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


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