Order on Order Types
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Date
2018-06
Publication Type
Journal Article
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yes
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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.
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published
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Journal / series
Volume
59 (4)
Pages / Article No.
886 - 922
Publisher
Springer
Event
Edition / version
Methods
Software
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Date collected
Date created
Subject
Point set; Order type; Planar graph
Organisational unit
03457 - Welzl, Emo (emeritus) / Welzl, Emo (emeritus)
Notes
It was possible to publish this article open access thanks to a Swiss National Licence with the publisher.