### 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-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 language | English |
---|---|

Pages (from-to) | 886-922 |

Number of pages | 37 |

Journal | Discrete and Computational Geometry |

Volume | 59 |

Issue number | 4 |

DOIs | |

Publication status | Published - 1 Jun 2018 |

Externally published | Yes |

### Fingerprint

### 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

*Discrete and Computational Geometry*,

*59*(4), 886-922. https://doi.org/10.1007/s00454-017-9912-9

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

Research output: Contribution to journal › Article › Research › peer-review

*Discrete and Computational Geometry*, vol. 59, no. 4, pp. 886-922. https://doi.org/10.1007/s00454-017-9912-9

}

TY - JOUR

T1 - Order on Order Types

AU - Pilz, Alexander

AU - Welzl, Emo

PY - 2018/6/1

Y1 - 2018/6/1

N2 - 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-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.

AB - 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-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.

KW - Order type

KW - Planar graph

KW - Point set

UR - http://www.scopus.com/inward/record.url?scp=85025630925&partnerID=8YFLogxK

U2 - 10.1007/s00454-017-9912-9

DO - 10.1007/s00454-017-9912-9

M3 - Article

VL - 59

SP - 886

EP - 922

JO - Discrete & computational geometry

JF - Discrete & computational geometry

SN - 0179-5376

IS - 4

ER -