Minimal representations of order types by geometric graphs

Oswin Aichholzer; Martin Balko; Michael Hoffmann; Jan Kynčl; Wolfgang Mulzer; Irene Parada; Alexander Pilz; Manfred Scheucher; Pavel Valtr; Birgit Vogtenhuber; Emo Welzl

doi:10.7155/jgaa.00545

Minimal representations of order types by geometric graphs

Oswin Aichholzer, Martin Balko, Michael Hoffmann, Jan Kynčl, Wolfgang Mulzer, Irene Parada, Alexander Pilz, Manfred Scheucher, Pavel Valtr, Birgit Vogtenhuber, Emo Welzl

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

Abstract

In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which prevent the order type from changing under continuous motion of vertices. That is, in the geometric graph on S whose edges are the exit edges, in order to change the order type of S, at least one vertex needs to move across an exit edge. Exit edges have a natural dual characterization, which allows us to efficiently compute them and to bound their number.

Original language	English
Pages (from-to)	551-572
Number of pages	22
Journal	Journal of Graph Algorithms and Applications
Volume	24
Issue number	4
DOIs	https://doi.org/10.7155/jgaa.00545
State	Published - 1 Jan 2020
Externally published	Yes

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science (all)
Computer Science Applications
Geometry and Topology
Computational Theory and Mathematics

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10.7155/jgaa.00545

Cite this

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title = "Minimal representations of order types by geometric graphs",

abstract = "In order to have a compact visualization of the order type of a given point set S, we are interested in geometric graphs on S with few edges that unambiguously display the order type of S. We introduce the concept of exit edges, which prevent the order type from changing under continuous motion of vertices. That is, in the geometric graph on S whose edges are the exit edges, in order to change the order type of S, at least one vertex needs to move across an exit edge. Exit edges have a natural dual characterization, which allows us to efficiently compute them and to bound their number.",

author = "Oswin Aichholzer and Martin Balko and Michael Hoffmann and Jan Kyn{\v c}l and Wolfgang Mulzer and Irene Parada and Alexander Pilz and Manfred Scheucher and Pavel Valtr and Birgit Vogtenhuber and Emo Welzl",

note = "Funding Information: This research is supported by the German Science Foundation (DFG), the Austrian Science Fund (FWF), and the Swiss National Science Foundation (SNSF) within the collaborative DACH project Arrangements and Drawings. O.A., I.P., and B.V. were supported by Austrian Science Fund (FWF) grant W1230 and I 3340-N35. M.B., J.K., and P.V. were supported by grant no. 18-19158S of the Czech Science Foundation (GAƒR). M.B. and J.K. were supported by Charles University project UNCE/SCI/004. M.B. has received funding from European Research Council (ERC) under the European Union's Horizon 2020 research. M.H. and E.W. were supported by SNSF Project 200021E-171681. A.P. was supported by a Schr{\"o}dinger fellowship of the Austrian Science Fund (FWF): J-3847-N35. M.S. was partially supported by DFG Grant FE 340/12-1. W.M. was partially supported by ERC StG 757609 and DFG Grant 3501/3-1. Publisher Copyright: {\textcopyright} 2020, Brown University. All rights reserved.",

year = "2020",

month = jan,

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doi = "10.7155/jgaa.00545",

language = "English",

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AU - Aichholzer, Oswin

AU - Balko, Martin

AU - Hoffmann, Michael

AU - Kynčl, Jan

AU - Mulzer, Wolfgang

AU - Parada, Irene

AU - Pilz, Alexander

AU - Scheucher, Manfred

AU - Valtr, Pavel

AU - Vogtenhuber, Birgit

AU - Welzl, Emo

N1 - Funding Information: This research is supported by the German Science Foundation (DFG), the Austrian Science Fund (FWF), and the Swiss National Science Foundation (SNSF) within the collaborative DACH project Arrangements and Drawings. O.A., I.P., and B.V. were supported by Austrian Science Fund (FWF) grant W1230 and I 3340-N35. M.B., J.K., and P.V. were supported by grant no. 18-19158S of the Czech Science Foundation (GAƒR). M.B. and J.K. were supported by Charles University project UNCE/SCI/004. M.B. has received funding from European Research Council (ERC) under the European Union's Horizon 2020 research. M.H. and E.W. were supported by SNSF Project 200021E-171681. A.P. was supported by a Schrödinger fellowship of the Austrian Science Fund (FWF): J-3847-N35. M.S. was partially supported by DFG Grant FE 340/12-1. W.M. was partially supported by ERC StG 757609 and DFG Grant 3501/3-1. Publisher Copyright: © 2020, Brown University. All rights reserved.

PY - 2020/1/1

Y1 - 2020/1/1

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Minimal representations of order types by geometric graphs

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