### 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 |
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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 |

### 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

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## Cite this

*Discrete and Computational Geometry*,

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