---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20456'
abstract:
- lang: eng
text: Given a locally finite set A⊆Rd and a coloring χ:A→{0,1,…,s}, we introduce
the chromatic Delaunay mosaic of χ, which is a Delaunay mosaic in Rs+d that represents
how points of different colors mingle. Our main results are bounds on the size
of the chromatic Delaunay mosaic, in which we assume that d and s are constants.
For example, if A is finite with n=#A, and the coloring is random, then the chromatic
Delaunay mosaic has O(n⌈d/2⌉) cells in expectation. In contrast, for Delone sets
and Poisson point processes in Rd, the expected number of cells within a closed
ball is only a constant times the number of points in this ball. Furthermore,
in R2 all colorings of a dense set of n points have chromatic Delaunay mosaics
of size O(n). This encourages the use of chromatic Delaunay mosaics in applications.
acknowledgement: The fourth author thanks Boris Aronov for insightful discussions
on the size of the overlay of Voronoi tessellations. Open access funding provided
by Institute of Science and Technology (IST Austria). This project has received
funding from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme, grant no. 788183, from the Wittgenstein
Prize, Austrian Science Fund (FWF), grant no. Z 342-N31, and from the DFG Collaborative
Research Center TRR 109, ‘Discretization in Geometry and Dynamics’, Austrian Science
Fund (FWF), grant no. I 02979-N35.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera di Montesano
orcid: 0000-0001-6249-0832
- first_name: Ondrej
full_name: Draganov, Ondrej
id: 2B23F01E-F248-11E8-B48F-1D18A9856A87
last_name: Draganov
orcid: 0000-0003-0464-3823
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M.
On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry.
2026;75:24-47. doi:10.1007/s00454-025-00778-7
apa: Biswas, R., Cultrera di Montesano, S., Draganov, O., Edelsbrunner, H., &
Saghafian, M. (2026). On the size of chromatic Delaunay mosaics. Discrete and
Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00778-7
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Ondrej Draganov, Herbert
Edelsbrunner, and Morteza Saghafian. “On the Size of Chromatic Delaunay Mosaics.”
Discrete and Computational Geometry. Springer Nature, 2026. https://doi.org/10.1007/s00454-025-00778-7.
ieee: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, and M.
Saghafian, “On the size of chromatic Delaunay mosaics,” Discrete and Computational
Geometry, vol. 75. Springer Nature, pp. 24–47, 2026.
ista: Biswas R, Cultrera di Montesano S, Draganov O, Edelsbrunner H, Saghafian M.
2026. On the size of chromatic Delaunay mosaics. Discrete and Computational Geometry.
75, 24–47.
mla: Biswas, Ranita, et al. “On the Size of Chromatic Delaunay Mosaics.” Discrete
and Computational Geometry, vol. 75, Springer Nature, 2026, pp. 24–47, doi:10.1007/s00454-025-00778-7.
short: R. Biswas, S. Cultrera di Montesano, O. Draganov, H. Edelsbrunner, M. Saghafian,
Discrete and Computational Geometry 75 (2026) 24–47.
corr_author: '1'
date_created: 2025-10-12T22:01:26Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-01-05T13:21:56Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-025-00778-7
ec_funded: 1
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oa_version: Published Version
page: 24-47
project:
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call_identifier: H2020
grant_number: '788183'
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publication: Discrete and Computational Geometry
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issn:
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publisher: Springer Nature
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title: On the size of chromatic Delaunay mosaics
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user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
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abstract:
- lang: eng
text: "An eight-partition of a finite set of points (respectively, of a continuous
mass distribution) in R^3\r\n consists of three planes that divide the space into
8 octants, such that each open octant contains at most 1/8 of the points (respectively,
of the mass). In 1966, Hadwiger showed that any mass distribution in R^3 admits
an eight-partition; moreover, one can prescribe the normal direction of one of
the three planes. The analogous result for finite point sets follows by a standard
limit argument. We prove the following variant of this result: any mass distribution
(or point set) in R^3 admits an eight-partition for which the intersection of
two of the planes is a line with a prescribed direction. Moreover, we present
an efficient algorithm for calculating an eight-partition of a set of n points
in R^3 (with prescribed normal direction of one of the planes) in time O(n^7/3).
A preliminary version of this work appeared in SoCG’24 (Aronov et al., 40th International
Symposium on Computational Geometry, 2024)."
acknowledgement: Work by BA was supported by NSF grants CCF 15-40656 and CCF 20-08551,
and by grant 2014/170 from the US-Israel Binational Science Foundation. Part of
this research was conducted while BA was visiting ISTA in the summers of 2022 and
2023. The visit of BA to ISTA in the summer of 2022 was supported by an ISTA Visiting
Professorship. Research of BA also partially supported by ERC grant no. 882971,
“GeoScape,” and by the Erdős Center. Work by AB was supported by Australian Research
Council grant DP220102212. Work by IR was supported by a Tandon School of Engineering
Fellowship and by NSF Grant CCF-20-08551. BA and AB would like to thank William
Steiger for insightful initial discussions of the problems addressed in this work.
Open Access funding enabled and organized by CAUL and its Member Institutions.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Boris
full_name: Aronov, Boris
last_name: Aronov
- first_name: Abdul
full_name: Basit, Abdul
last_name: Basit
- first_name: Indu
full_name: Ramesh, Indu
last_name: Ramesh
- first_name: Gianluca
full_name: Tasinato, Gianluca
id: 0433290C-AF8F-11E9-A4C7-F729E6697425
last_name: Tasinato
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Aronov B, Basit A, Ramesh I, Tasinato G, Wagner U. Eight-partitioning points
in 3D, and efficiently too. Discrete & Computational Geometry. 2026;75:1331-1355.
doi:10.1007/s00454-025-00739-0
apa: Aronov, B., Basit, A., Ramesh, I., Tasinato, G., & Wagner, U. (2026). Eight-partitioning
points in 3D, and efficiently too. Discrete & Computational Geometry.
Springer Nature. https://doi.org/10.1007/s00454-025-00739-0
chicago: Aronov, Boris, Abdul Basit, Indu Ramesh, Gianluca Tasinato, and Uli Wagner.
“Eight-Partitioning Points in 3D, and Efficiently Too.” Discrete & Computational
Geometry. Springer Nature, 2026. https://doi.org/10.1007/s00454-025-00739-0.
ieee: B. Aronov, A. Basit, I. Ramesh, G. Tasinato, and U. Wagner, “Eight-partitioning
points in 3D, and efficiently too,” Discrete & Computational Geometry,
vol. 75. Springer Nature, pp. 1331–1355, 2026.
ista: Aronov B, Basit A, Ramesh I, Tasinato G, Wagner U. 2026. Eight-partitioning
points in 3D, and efficiently too. Discrete & Computational Geometry. 75,
1331–1355.
mla: Aronov, Boris, et al. “Eight-Partitioning Points in 3D, and Efficiently Too.”
Discrete & Computational Geometry, vol. 75, Springer Nature, 2026,
pp. 1331–55, doi:10.1007/s00454-025-00739-0.
short: B. Aronov, A. Basit, I. Ramesh, G. Tasinato, U. Wagner, Discrete & Computational
Geometry 75 (2026) 1331–1355.
das_tickbox: '0'
date_created: 2025-06-22T22:02:07Z
date_published: 2026-06-01T00:00:00Z
date_updated: 2026-07-23T11:14:46Z
day: '01'
ddc:
- '500'
department:
- _id: UlWa
doi: 10.1007/s00454-025-00739-0
external_id:
arxiv:
- '2403.02627'
isi:
- '001506904300001'
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checksum: a32774a0d14f46cafbd77bfb9d9ac4b6
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related_material:
link:
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url: https://doi.org/10.1007/s00454-025-00759-w
record:
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supplementarymaterial: yes
title: Eight-partitioning points in 3D, and efficiently too
tmp:
image: /images/cc_by.png
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name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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volume: 75
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...
---
OA_place: publisher
OA_type: hybrid
_id: '17149'
abstract:
- lang: eng
text: The approximation of a circle with the edges of a fine square grid distorts
the perimeter by a factor about 4/Pi. We prove that this factor is the same on
average (in the ergodic sense) for approximations of any rectifiable curve by
the edges of any non-exotic Delaunay mosaic (known as Voronoi path), and extend
the results to all dimensions, generalizing Voronoi paths to Voronoi scapes.
acknowledgement: "The authors thank Ranita Biswas and Tatiana Ezubova for the collaboration
on computational experiments that motivated the work reported in this paper. The
authors also thank Daniel Bonnema for proofreading and noticing an issue with the
original proof of Lemma 4.3.\r\nOpen access funding provided by Institute of Science
and Technology (IST Austria).\r\nThis project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme, Grant No. 788183, from the Wittgenstein Prize, Austrian Science Fund
(FWF), Grant No. Z 342-N31, and from the DFG Collaborative Research Center TRR 109,
‘Discretization in Geometry and Dynamics’, Austrian Science Fund (FWF), Grant No.
I 02979-N35."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Anton
full_name: Nikitenko, Anton
id: 3E4FF1BA-F248-11E8-B48F-1D18A9856A87
last_name: Nikitenko
orcid: 0000-0002-0659-3201
citation:
ama: Edelsbrunner H, Nikitenko A. Average and expected distortion of Voronoi paths
and scapes. Discrete & Computational Geometry. 2025;73:490-499. doi:10.1007/s00454-024-00660-y
apa: Edelsbrunner, H., & Nikitenko, A. (2025). Average and expected distortion
of Voronoi paths and scapes. Discrete & Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-024-00660-y
chicago: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion
of Voronoi Paths and Scapes.” Discrete & Computational Geometry. Springer
Nature, 2025. https://doi.org/10.1007/s00454-024-00660-y.
ieee: H. Edelsbrunner and A. Nikitenko, “Average and expected distortion of Voronoi
paths and scapes,” Discrete & Computational Geometry, vol. 73. Springer
Nature, pp. 490–499, 2025.
ista: Edelsbrunner H, Nikitenko A. 2025. Average and expected distortion of Voronoi
paths and scapes. Discrete & Computational Geometry. 73, 490–499.
mla: Edelsbrunner, Herbert, and Anton Nikitenko. “Average and Expected Distortion
of Voronoi Paths and Scapes.” Discrete & Computational Geometry, vol.
73, Springer Nature, 2025, pp. 490–99, doi:10.1007/s00454-024-00660-y.
short: H. Edelsbrunner, A. Nikitenko, Discrete & Computational Geometry 73 (2025)
490–499.
corr_author: '1'
date_created: 2024-06-16T22:01:07Z
date_published: 2025-03-01T00:00:00Z
date_updated: 2026-02-16T12:18:50Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-024-00660-y
ec_funded: 1
external_id:
arxiv:
- '2012.03350'
isi:
- '001238566200004'
pmid:
- '39974750'
file:
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checksum: ffb0c818222138f9f113f4bbea41e834
content_type: application/pdf
creator: dernst
date_created: 2025-04-23T07:31:32Z
date_updated: 2025-04-23T07:31:32Z
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file_size: 283443
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language:
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oa: 1
oa_version: Published Version
page: 490-499
pmid: 1
project:
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call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: Mathematics, Computer Science
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Average and expected distortion of Voronoi paths and scapes
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 73
year: '2025'
...
---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '20657'
abstract:
- lang: eng
text: 'The Upper Bound Theorem for convex polytopes implies that the p-th Betti
number of the Čech complex of any set of N points in ℝ^d and any radius satisfies
β_p = O(N^m), with m = min{p+1, ⌈d/2⌉}. We construct sets in even and odd dimensions,
which prove that this upper bound is asymptotically tight. For example, we describe
a set of N = 2(n+1) points in ℝ³ and two radii such that the first Betti number
of the Čech complex at one radius is (n+1)² - 1, and the second Betti number of
the Čech complex at the other radius is n². '
acknowledgement: The first author is supported by the European Research Council (ERC),
grant no. 788183, and by the DFG Collaborative Research Center TRR 109, Austrian
Science Fund (FWF), grant no. I 02979-N35. The second author is supported by the
European Research Council (ERC), grant “GeoScape” and by the Hungarian Science Foundation
(NKFIH), grant K-131529. Both authors are supported by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant no. Z 342-N31.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
citation:
ama: Edelsbrunner H, Pach J. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry. 2025. doi:10.1007/s00454-025-00796-5
apa: Edelsbrunner, H., & Pach, J. (2025). Maximum Betti numbers of Čech complexes.
Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-025-00796-5
chicago: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry. Springer Nature, 2025. https://doi.org/10.1007/s00454-025-00796-5.
ieee: H. Edelsbrunner and J. Pach, “Maximum Betti numbers of Čech complexes,” Discrete
& Computational Geometry. Springer Nature, 2025.
ista: Edelsbrunner H, Pach J. 2025. Maximum Betti numbers of Čech complexes. Discrete
& Computational Geometry.
mla: Edelsbrunner, Herbert, and János Pach. “Maximum Betti Numbers of Čech Complexes.”
Discrete & Computational Geometry, Springer Nature, 2025, doi:10.1007/s00454-025-00796-5.
short: H. Edelsbrunner, J. Pach, Discrete & Computational Geometry (2025).
corr_author: '1'
date_created: 2025-11-19T09:44:58Z
date_published: 2025-11-10T00:00:00Z
date_updated: 2025-12-01T15:19:21Z
day: '10'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-025-00796-5
ec_funded: 1
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arxiv:
- '2310.14801'
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- '001610592600001'
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isi: 1
language:
- iso: eng
main_file_link:
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url: https://doi.org/10.1007/s00454-025-00796-5
month: '11'
oa: 1
oa_version: Published Version
project:
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call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
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title: Maximum Betti numbers of Čech complexes
tmp:
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type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2025'
...
---
OA_place: repository
OA_type: green
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abstract:
- lang: eng
text: The Tverberg theorem is one of the cornerstones of discrete geometry. It states
that, given a set X of at least (d+1)(r−1)+1 points in Rd, one can find a partition
X=X1∪⋯∪Xr of X, such that the convex hulls of the Xi, i=1,…,r, all share a common
point. In this paper, we prove a trengthening of this theorem that guarantees
a partition which, in addition to the above, has the property that the boundaries
of full-dimensional convex hulls have pairwise nonempty intersections. Possible
generalizations and algorithmic aspects are also discussed. As a concrete application,
we show that any n points in the plane in general position span ⌊n/3⌋ vertex-disjoint
triangles that are pairwise crossing, meaning that their boundaries have pairwise
nonempty intersections; this number is clearly best possible. A previous result
of Álvarez-Rebollar et al. guarantees ⌊n/6⌋pairwise crossing triangles. Our result
generalizes to a result about simplices in Rd, d≥2.
acknowledgement: "Part of the research leading to this paper was done during the 16th
Gremo Workshop on Open Problems (GWOP), Waltensburg, Switzerland, June 12–16, 2018.
We thank Patrick Schnider for suggesting the problem, and Stefan Felsner, Malte
Milatz, and Emo Welzl for fruitful discussions during the workshop. We also thank
Stefan Felsner and Manfred Scheucher for finding, communicating the example from
Sect. 3.3, and the kind permission to include their visualization of the point set.
We thank Dömötör Pálvölgyi, the SoCG reviewers, and DCG reviewers for various helpful
comments.\r\nR. Fulek gratefully acknowledges support from Austrian Science Fund
(FWF), Project M2281-N35. A. Kupavskii was supported by the Advanced Postdoc.Mobility
Grant no. P300P2_177839 of the Swiss National Science Foundation. Research by P.
Valtr was supported by the Grant no. 18-19158 S of the Czech Science Foundation
(GAČR)."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Radoslav
full_name: Fulek, Radoslav
id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87
last_name: Fulek
orcid: 0000-0001-8485-1774
- first_name: Bernd
full_name: Gärtner, Bernd
last_name: Gärtner
- first_name: Andrey
full_name: Kupavskii, Andrey
last_name: Kupavskii
- first_name: Pavel
full_name: Valtr, Pavel
last_name: Valtr
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. The crossing Tverberg theorem.
Discrete and Computational Geometry. 2024;72:831-848. doi:10.1007/s00454-023-00532-x
apa: Fulek, R., Gärtner, B., Kupavskii, A., Valtr, P., & Wagner, U. (2024).
The crossing Tverberg theorem. Discrete and Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-023-00532-x
chicago: Fulek, Radoslav, Bernd Gärtner, Andrey Kupavskii, Pavel Valtr, and Uli
Wagner. “The Crossing Tverberg Theorem.” Discrete and Computational Geometry.
Springer Nature, 2024. https://doi.org/10.1007/s00454-023-00532-x.
ieee: R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, and U. Wagner, “The crossing
Tverberg theorem,” Discrete and Computational Geometry, vol. 72. Springer
Nature, pp. 831–848, 2024.
ista: Fulek R, Gärtner B, Kupavskii A, Valtr P, Wagner U. 2024. The crossing Tverberg
theorem. Discrete and Computational Geometry. 72, 831–848.
mla: Fulek, Radoslav, et al. “The Crossing Tverberg Theorem.” Discrete and Computational
Geometry, vol. 72, Springer Nature, 2024, pp. 831–48, doi:10.1007/s00454-023-00532-x.
short: R. Fulek, B. Gärtner, A. Kupavskii, P. Valtr, U. Wagner, Discrete and Computational
Geometry 72 (2024) 831–848.
date_created: 2023-08-06T22:01:12Z
date_published: 2024-09-01T00:00:00Z
date_updated: 2025-04-14T13:52:36Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-023-00532-x
external_id:
arxiv:
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isi:
- '001038546500001'
intvolume: ' 72'
isi: 1
language:
- iso: eng
main_file_link:
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url: https://doi.org/10.48550/arXiv.1812.04911
month: '09'
oa: 1
oa_version: Preprint
page: 831-848
project:
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call_identifier: FWF
grant_number: M02281
name: Eliminating intersections in drawings of graphs
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
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issn:
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publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
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scopus_import: '1'
status: public
title: The crossing Tverberg theorem
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 72
year: '2024'
...
---
_id: '14345'
abstract:
- lang: eng
text: For a locally finite set in R2, the order-k Brillouin tessellations form an
infinite sequence of convex face-to-face tilings of the plane. If the set is coarsely
dense and generic, then the corresponding infinite sequences of minimum and maximum
angles are both monotonic in k. As an example, a stationary Poisson point process
in R2 is locally finite, coarsely dense, and generic with probability one. For
such a set, the distributions of angles in the Voronoi tessellations, Delaunay
mosaics, and Brillouin tessellations are independent of the order and can be derived
from the formula for angles in order-1 Delaunay mosaics given by Miles (Math.
Biosci. 6, 85–127 (1970)).
acknowledgement: Work by all authors but A. Garber is supported by the European Research
Council (ERC), Grant No. 788183, by the Wittgenstein Prize, Austrian Science Fund
(FWF), Grant No. Z 342-N31, and by the DFG Collaborative Research Center TRR 109,
Austrian Science Fund (FWF), Grant No. I 02979-N35. Work by A. Garber is partially
supported by the Alexander von Humboldt Foundation.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Alexey
full_name: Garber, Alexey
last_name: Garber
- first_name: Mohadese
full_name: Ghafari, Mohadese
last_name: Ghafari
- first_name: Teresa
full_name: Heiss, Teresa
id: 4879BB4E-F248-11E8-B48F-1D18A9856A87
last_name: Heiss
orcid: 0000-0002-1780-2689
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. On angles in higher
order Brillouin tessellations and related tilings in the plane. Discrete and
Computational Geometry. 2024;72:29-48. doi:10.1007/s00454-023-00566-1
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2024).
On angles in higher order Brillouin tessellations and related tilings in the plane.
Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00566-1
chicago: Edelsbrunner, Herbert, Alexey Garber, Mohadese Ghafari, Teresa Heiss, and
Morteza Saghafian. “On Angles in Higher Order Brillouin Tessellations and Related
Tilings in the Plane.” Discrete and Computational Geometry. Springer Nature,
2024. https://doi.org/10.1007/s00454-023-00566-1.
ieee: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, and M. Saghafian, “On angles
in higher order Brillouin tessellations and related tilings in the plane,” Discrete
and Computational Geometry, vol. 72. Springer Nature, pp. 29–48, 2024.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2024. On angles
in higher order Brillouin tessellations and related tilings in the plane. Discrete
and Computational Geometry. 72, 29–48.
mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations
and Related Tilings in the Plane.” Discrete and Computational Geometry,
vol. 72, Springer Nature, 2024, pp. 29–48, doi:10.1007/s00454-023-00566-1.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete
and Computational Geometry 72 (2024) 29–48.
corr_author: '1'
date_created: 2023-09-17T22:01:10Z
date_published: 2024-07-01T00:00:00Z
date_updated: 2025-04-23T08:41:59Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-023-00566-1
ec_funded: 1
external_id:
arxiv:
- '2204.01076'
isi:
- '001060727600004'
pmid:
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oa: 1
oa_version: Published Version
page: 29-48
pmid: 1
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call_identifier: H2020
grant_number: '788183'
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publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
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publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On angles in higher order Brillouin tessellations and related tilings in the
plane
tmp:
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legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 72
year: '2024'
...
---
_id: '11999'
abstract:
- lang: eng
text: 'A simple drawing D(G) of a graph G is one where each pair of edges share
at most one point: either a common endpoint or a proper crossing. An edge e in
the complement of G can be inserted into D(G) if there exists a simple drawing
of G+e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is
rectilinear (pseudolinear), that is, the edges can be extended into an arrangement
of lines (pseudolines), then any edge in the complement of G can be inserted.
In contrast, we show that it is NP-complete to decide whether one edge can be
inserted into a simple drawing. This remains true even if we assume that the drawing
is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles.
On the positive side, we show that, given an arrangement of pseudocircles A and
a pseudosegment σ, it can be decided in polynomial time whether there exists a
pseudocircle Φσ extending σ for which A∪{Φσ} is again an arrangement of pseudocircles.'
acknowledgement: 'This work was started during the 6th Austrian–Japanese–Mexican–Spanish
Workshop on Discrete Geometry in June 2019 in Austria. We thank all the participants
for the good atmosphere as well as discussions on the topic. Also, we thank Jan
Kynčl for sending us remarks on a preliminary version of this work and an anonymous
referee for further helpful comments.Alan Arroyo was funded by the Marie Skłodowska-Curie
grant agreement No 754411. Fabian Klute was partially supported by the Netherlands
Organisation for Scientific Research (NWO) under project no. 612.001.651 and by
the Austrian Science Fund (FWF): J-4510. Irene Parada and Birgit Vogtenhuber were
partially supported by the Austrian Science Fund (FWF): W1230 and within the collaborative
DACH project Arrangements and Drawings as FWF project I 3340-N35. Irene Parada was
also partially supported by the Independent Research Fund Denmark grant 2020-2023
(9131-00044B) Dynamic Network Analysis and by the Margarita Salas Fellowship funded
by the Ministry of Universities of Spain and the European Union (NextGenerationEU).
Tilo Wiedera was supported by the German Research Foundation (DFG) grant CH 897/2-2.'
article_processing_charge: Yes (in subscription journal)
article_type: original
arxiv: 1
author:
- first_name: Alan M
full_name: Arroyo Guevara, Alan M
id: 3207FDC6-F248-11E8-B48F-1D18A9856A87
last_name: Arroyo Guevara
orcid: 0000-0003-2401-8670
- first_name: Fabian
full_name: Klute, Fabian
last_name: Klute
- first_name: Irene
full_name: Parada, Irene
last_name: Parada
- first_name: Birgit
full_name: Vogtenhuber, Birgit
last_name: Vogtenhuber
- first_name: Raimund
full_name: Seidel, Raimund
last_name: Seidel
- first_name: Tilo
full_name: Wiedera, Tilo
last_name: Wiedera
citation:
ama: Arroyo Guevara AM, Klute F, Parada I, Vogtenhuber B, Seidel R, Wiedera T. Inserting
one edge into a simple drawing is hard. Discrete and Computational Geometry.
2023;69:745–770. doi:10.1007/s00454-022-00394-9
apa: Arroyo Guevara, A. M., Klute, F., Parada, I., Vogtenhuber, B., Seidel, R.,
& Wiedera, T. (2023). Inserting one edge into a simple drawing is hard. Discrete
and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00394-9
chicago: Arroyo Guevara, Alan M, Fabian Klute, Irene Parada, Birgit Vogtenhuber,
Raimund Seidel, and Tilo Wiedera. “Inserting One Edge into a Simple Drawing Is
Hard.” Discrete and Computational Geometry. Springer Nature, 2023. https://doi.org/10.1007/s00454-022-00394-9.
ieee: A. M. Arroyo Guevara, F. Klute, I. Parada, B. Vogtenhuber, R. Seidel, and
T. Wiedera, “Inserting one edge into a simple drawing is hard,” Discrete and
Computational Geometry, vol. 69. Springer Nature, pp. 745–770, 2023.
ista: Arroyo Guevara AM, Klute F, Parada I, Vogtenhuber B, Seidel R, Wiedera T.
2023. Inserting one edge into a simple drawing is hard. Discrete and Computational
Geometry. 69, 745–770.
mla: Arroyo Guevara, Alan M., et al. “Inserting One Edge into a Simple Drawing Is
Hard.” Discrete and Computational Geometry, vol. 69, Springer Nature, 2023,
pp. 745–770, doi:10.1007/s00454-022-00394-9.
short: A.M. Arroyo Guevara, F. Klute, I. Parada, B. Vogtenhuber, R. Seidel, T. Wiedera,
Discrete and Computational Geometry 69 (2023) 745–770.
date_created: 2022-08-28T22:02:01Z
date_published: 2023-04-01T00:00:00Z
date_updated: 2025-04-14T07:43:59Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/s00454-022-00394-9
ec_funded: 1
external_id:
arxiv:
- '1909.07347'
isi:
- '000840292800001'
file:
- access_level: open_access
checksum: def7ae3b28d9fd6aec16450e40090302
content_type: application/pdf
creator: alisjak
date_created: 2022-08-29T11:23:15Z
date_updated: 2022-08-29T11:23:15Z
file_id: '12006'
file_name: 2022_DiscreteandComputionalGeometry_Arroyo.pdf
file_size: 1002218
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success: 1
file_date_updated: 2022-08-29T11:23:15Z
has_accepted_license: '1'
intvolume: ' 69'
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language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 745–770
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Inserting one edge into a simple drawing is hard
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 69
year: '2023'
...
---
_id: '12287'
abstract:
- lang: eng
text: We present criteria for establishing a triangulation of a manifold. Given
a manifold M, a simplicial complex A, and a map H from the underlying space of
A to M, our criteria are presented in local coordinate charts for M, and ensure
that H is a homeomorphism. These criteria do not require a differentiable structure,
or even an explicit metric on M. No Delaunay property of A is assumed. The result
provides a triangulation guarantee for algorithms that construct a simplicial
complex by working in local coordinate patches. Because the criteria are easily
verified in such a setting, they are expected to be of general use.
acknowledgement: "This work has been funded by the European Research Council under
the European Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations
of Geometric Understanding in Higher Dimensions). Arijit Ghosh is supported by Ramanujan
Fellowship (No. SB/S2/RJN-064/2015). Part of this work was done when Arijit Ghosh
was a Researcher at Max-Planck-Institute for Informatics, Germany, supported by
the IndoGerman Max Planck Center for Computer Science (IMPECS). Mathijs Wintraecken
also received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska-Curie grant agreement No. 754411 and the Austrian
Science Fund (FWF): M-3073. A part of the results described in this paper were presented
at SoCG 2018 and in [3]. \r\nOpen access funding provided by the Austrian Science
Fund (FWF)."
article_processing_charge: No
article_type: original
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- first_name: Ramsay
full_name: Dyer, Ramsay
last_name: Dyer
- first_name: Arijit
full_name: Ghosh, Arijit
last_name: Ghosh
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat J-D, Dyer R, Ghosh A, Wintraecken M. Local criteria for triangulating
general manifolds. Discrete & Computational Geometry. 2023;69:156-191.
doi:10.1007/s00454-022-00431-7
apa: Boissonnat, J.-D., Dyer, R., Ghosh, A., & Wintraecken, M. (2023). Local
criteria for triangulating general manifolds. Discrete & Computational
Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00431-7
chicago: Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, and Mathijs Wintraecken.
“Local Criteria for Triangulating General Manifolds.” Discrete & Computational
Geometry. Springer Nature, 2023. https://doi.org/10.1007/s00454-022-00431-7.
ieee: J.-D. Boissonnat, R. Dyer, A. Ghosh, and M. Wintraecken, “Local criteria for
triangulating general manifolds,” Discrete & Computational Geometry,
vol. 69. Springer Nature, pp. 156–191, 2023.
ista: Boissonnat J-D, Dyer R, Ghosh A, Wintraecken M. 2023. Local criteria for triangulating
general manifolds. Discrete & Computational Geometry. 69, 156–191.
mla: Boissonnat, Jean-Daniel, et al. “Local Criteria for Triangulating General Manifolds.”
Discrete & Computational Geometry, vol. 69, Springer Nature, 2023,
pp. 156–91, doi:10.1007/s00454-022-00431-7.
short: J.-D. Boissonnat, R. Dyer, A. Ghosh, M. Wintraecken, Discrete & Computational
Geometry 69 (2023) 156–191.
corr_author: '1'
date_created: 2023-01-16T10:04:06Z
date_published: 2023-01-01T00:00:00Z
date_updated: 2025-04-14T07:44:00Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00431-7
ec_funded: 1
external_id:
isi:
- '000862193600001'
file:
- access_level: open_access
checksum: 46352e0ee71e460848f88685ca852681
content_type: application/pdf
creator: dernst
date_created: 2023-02-02T11:01:10Z
date_updated: 2023-02-02T11:01:10Z
file_id: '12488'
file_name: 2023_DiscreteCompGeometry_Boissonnat.pdf
file_size: 582850
relation: main_file
success: 1
file_date_updated: 2023-02-02T11:01:10Z
has_accepted_license: '1'
intvolume: ' 69'
isi: 1
keyword:
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Theoretical Computer Science
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 156-191
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: fc390959-9c52-11eb-aca3-afa58bd282b2
grant_number: M03073
name: Learning and triangulating manifolds via collapses
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Local criteria for triangulating general manifolds
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 69
year: '2023'
...
---
_id: '12709'
abstract:
- lang: eng
text: Given a finite set A ⊂ ℝ^d, let Cov_{r,k} denote the set of all points within
distance r to at least k points of A. Allowing r and k to vary, we obtain a 2-parameter
family of spaces that grow larger when r increases or k decreases, called the
multicover bifiltration. Motivated by the problem of computing the homology of
this bifiltration, we introduce two closely related combinatorial bifiltrations,
one polyhedral and the other simplicial, which are both topologically equivalent
to the multicover bifiltration and far smaller than a Čech-based model considered
in prior work of Sheehy. Our polyhedral construction is a bifiltration of the
rhomboid tiling of Edelsbrunner and Osang, and can be efficiently computed using
a variant of an algorithm given by these authors as well. Using an implementation
for dimension 2 and 3, we provide experimental results. Our simplicial construction
is useful for understanding the polyhedral construction and proving its correctness.
acknowledgement: We thank the anonymous reviewers for many helpful comments and suggestions,
which led to substantial improvements of the paper. The first two authors were supported
by the Austrian Science Fund (FWF) grant number P 29984-N35 and W1230. The first
author was partly supported by an Austrian Marshall Plan Scholarship, and by the
Brummer & Partners MathDataLab. A conference version of this paper was presented
at the 37th International Symposium on Computational Geometry (SoCG 2021). Open
access funding provided by the Royal Institute of Technology.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: René
full_name: Corbet, René
last_name: Corbet
- first_name: Michael
full_name: Kerber, Michael
id: 36E4574A-F248-11E8-B48F-1D18A9856A87
last_name: Kerber
orcid: 0000-0002-8030-9299
- first_name: Michael
full_name: Lesnick, Michael
last_name: Lesnick
- first_name: Georg F
full_name: Osang, Georg F
id: 464B40D6-F248-11E8-B48F-1D18A9856A87
last_name: Osang
orcid: 0000-0002-8882-5116
citation:
ama: Corbet R, Kerber M, Lesnick M, Osang GF. Computing the multicover bifiltration.
Discrete and Computational Geometry. 2023;70:376-405. doi:10.1007/s00454-022-00476-8
apa: Corbet, R., Kerber, M., Lesnick, M., & Osang, G. F. (2023). Computing the
multicover bifiltration. Discrete and Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-022-00476-8
chicago: Corbet, René, Michael Kerber, Michael Lesnick, and Georg F Osang. “Computing
the Multicover Bifiltration.” Discrete and Computational Geometry. Springer
Nature, 2023. https://doi.org/10.1007/s00454-022-00476-8.
ieee: R. Corbet, M. Kerber, M. Lesnick, and G. F. Osang, “Computing the multicover
bifiltration,” Discrete and Computational Geometry, vol. 70. Springer Nature,
pp. 376–405, 2023.
ista: Corbet R, Kerber M, Lesnick M, Osang GF. 2023. Computing the multicover bifiltration.
Discrete and Computational Geometry. 70, 376–405.
mla: Corbet, René, et al. “Computing the Multicover Bifiltration.” Discrete and
Computational Geometry, vol. 70, Springer Nature, 2023, pp. 376–405, doi:10.1007/s00454-022-00476-8.
short: R. Corbet, M. Kerber, M. Lesnick, G.F. Osang, Discrete and Computational
Geometry 70 (2023) 376–405.
date_created: 2023-03-05T23:01:06Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2025-07-10T12:01:57Z
day: '01'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00476-8
external_id:
arxiv:
- '2103.07823'
isi:
- '000936496800001'
pmid:
- '37581017'
file:
- access_level: open_access
checksum: 71ce7e59f7ee4620acc704fecca620c2
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creator: cchlebak
date_created: 2023-03-07T14:40:14Z
date_updated: 2023-03-07T14:40:14Z
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intvolume: ' 70'
isi: 1
language:
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month: '09'
oa: 1
oa_version: Published Version
page: 376-405
pmid: 1
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '9605'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Computing the multicover bifiltration
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2023'
...
---
_id: '12764'
abstract:
- lang: eng
text: We study a new discretization of the Gaussian curvature for polyhedral surfaces.
This discrete Gaussian curvature is defined on each conical singularity of a polyhedral
surface as the quotient of the angle defect and the area of the Voronoi cell corresponding
to the singularity. We divide polyhedral surfaces into discrete conformal classes
using a generalization of discrete conformal equivalence pioneered by Feng Luo.
We subsequently show that, in every discrete conformal class, there exists a polyhedral
surface with constant discrete Gaussian curvature. We also provide explicit examples
to demonstrate that this surface is in general not unique.
acknowledgement: Open access funding provided by the Austrian Science Fund (FWF).
This research was supported by the FWF grant, Project number I4245-N35, and by the
Deutsche Forschungsgemeinschaft (DFG - German Research Foundation) - Project-ID
195170736 - TRR109.
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Hana
full_name: Kourimska, Hana
id: D9B8E14C-3C26-11EA-98F5-1F833DDC885E
last_name: Kourimska
orcid: 0000-0001-7841-0091
citation:
ama: Kourimska H. Discrete yamabe problem for polyhedral surfaces. Discrete and
Computational Geometry. 2023;70:123-153. doi:10.1007/s00454-023-00484-2
apa: Kourimska, H. (2023). Discrete yamabe problem for polyhedral surfaces. Discrete
and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00484-2
chicago: Kourimska, Hana. “Discrete Yamabe Problem for Polyhedral Surfaces.” Discrete
and Computational Geometry. Springer Nature, 2023. https://doi.org/10.1007/s00454-023-00484-2.
ieee: H. Kourimska, “Discrete yamabe problem for polyhedral surfaces,” Discrete
and Computational Geometry, vol. 70. Springer Nature, pp. 123–153, 2023.
ista: Kourimska H. 2023. Discrete yamabe problem for polyhedral surfaces. Discrete
and Computational Geometry. 70, 123–153.
mla: Kourimska, Hana. “Discrete Yamabe Problem for Polyhedral Surfaces.” Discrete
and Computational Geometry, vol. 70, Springer Nature, 2023, pp. 123–53, doi:10.1007/s00454-023-00484-2.
short: H. Kourimska, Discrete and Computational Geometry 70 (2023) 123–153.
corr_author: '1'
date_created: 2023-03-26T22:01:09Z
date_published: 2023-07-01T00:00:00Z
date_updated: 2025-04-23T08:59:15Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-023-00484-2
external_id:
isi:
- '000948148000001'
pmid:
- '37292248'
file:
- access_level: open_access
checksum: cdbf90ba4a7ddcb190d37b9e9d4cb9d3
content_type: application/pdf
creator: dernst
date_created: 2023-10-04T11:46:24Z
date_updated: 2023-10-04T11:46:24Z
file_id: '14396'
file_name: 2023_DiscreteGeometry_Kourimska.pdf
file_size: 1026683
relation: main_file
success: 1
file_date_updated: 2023-10-04T11:46:24Z
has_accepted_license: '1'
intvolume: ' 70'
isi: 1
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 123-153
pmid: 1
project:
- _id: 26AD5D90-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I04245
name: Algebraic Footprints of Geometric Features in Homology
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Discrete yamabe problem for polyhedral surfaces
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2023'
...
---
_id: '13270'
abstract:
- lang: eng
text: "Consider a geodesic triangle on a surface of constant curvature and subdivide
it recursively into four triangles by joining the midpoints of its edges. We show
the existence of a uniform δ>0\r\n such that, at any step of the subdivision,
all the triangle angles lie in the interval (δ,π−δ)\r\n. Additionally, we exhibit
stabilising behaviours for both angles and lengths as this subdivision progresses."
acknowledgement: Open access funding provided by the Institute of Science and Technology
(IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Florestan R
full_name: Brunck, Florestan R
id: 6ab6e556-f394-11eb-9cf6-9dfb78f00d8d
last_name: Brunck
citation:
ama: Brunck FR. Iterated medial triangle subdivision in surfaces of constant curvature.
Discrete and Computational Geometry. 2023;70(3):1059-1089. doi:10.1007/s00454-023-00500-5
apa: Brunck, F. R. (2023). Iterated medial triangle subdivision in surfaces of constant
curvature. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-023-00500-5
chicago: Brunck, Florestan R. “Iterated Medial Triangle Subdivision in Surfaces
of Constant Curvature.” Discrete and Computational Geometry. Springer Nature,
2023. https://doi.org/10.1007/s00454-023-00500-5.
ieee: F. R. Brunck, “Iterated medial triangle subdivision in surfaces of constant
curvature,” Discrete and Computational Geometry, vol. 70, no. 3. Springer
Nature, pp. 1059–1089, 2023.
ista: Brunck FR. 2023. Iterated medial triangle subdivision in surfaces of constant
curvature. Discrete and Computational Geometry. 70(3), 1059–1089.
mla: Brunck, Florestan R. “Iterated Medial Triangle Subdivision in Surfaces of Constant
Curvature.” Discrete and Computational Geometry, vol. 70, no. 3, Springer
Nature, 2023, pp. 1059–89, doi:10.1007/s00454-023-00500-5.
short: F.R. Brunck, Discrete and Computational Geometry 70 (2023) 1059–1089.
corr_author: '1'
date_created: 2023-07-23T22:01:14Z
date_published: 2023-07-05T00:00:00Z
date_updated: 2024-10-09T21:06:01Z
day: '05'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/s00454-023-00500-5
external_id:
arxiv:
- '2107.04112'
isi:
- '001023742800003'
file:
- access_level: open_access
checksum: 865e68daafdd4edcfc280172ec50f5ea
content_type: application/pdf
creator: dernst
date_created: 2024-01-29T11:15:22Z
date_updated: 2024-01-29T11:15:22Z
file_id: '14897'
file_name: 2023_DiscreteComputGeometry_Brunck.pdf
file_size: 1466020
relation: main_file
success: 1
file_date_updated: 2024-01-29T11:15:22Z
has_accepted_license: '1'
intvolume: ' 70'
isi: 1
issue: '3'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 1059-1089
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Iterated medial triangle subdivision in surfaces of constant curvature
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 70
year: '2023'
...
---
_id: '11593'
abstract:
- lang: eng
text: 'A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The Z2 -genus
of a graph G is the minimum g such that G has an independently even drawing on
the orientable surface of genus g. An unpublished result by Robertson and Seymour
implies that for every t, every graph of sufficiently large genus contains as
a minor a projective t×t grid or one of the following so-called t -Kuratowski
graphs: K3,t, or t copies of K5 or K3,3 sharing at most two common vertices. We
show that the Z2-genus of graphs in these families is unbounded in t; in fact,
equal to their genus. Together, this implies that the genus of a graph is bounded
from above by a function of its Z2-genus, solving a problem posed by Schaefer
and Štefankovič, and giving an approximate version of the Hanani–Tutte theorem
on orientable surfaces. We also obtain an analogous result for Euler genus and
Euler Z2-genus of graphs.'
acknowledgement: "We thank Zdeněk Dvořák, Xavier Goaoc, and Pavel Paták for helpful
discussions. We also thank Bojan Mohar, Paul Seymour, Gelasio Salazar, Jim Geelen,
and John Maharry for information about their unpublished results related to Conjecture
3.1. Finally we thank the reviewers for corrections and suggestions for improving
the presentation.\r\nSupported by Austrian Science Fund (FWF): M2281-N35. Supported
by project 19-04113Y of the Czech Science Foundation (GAČR), by the Czech-French
collaboration project EMBEDS II (CZ: 7AMB17FR029, FR: 38087RM), and by Charles University
project UNCE/SCI/004."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Radoslav
full_name: Fulek, Radoslav
id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87
last_name: Fulek
orcid: 0000-0001-8485-1774
- first_name: Jan
full_name: Kynčl, Jan
last_name: Kynčl
citation:
ama: Fulek R, Kynčl J. The Z2-Genus of Kuratowski minors. Discrete and Computational
Geometry. 2022;68:425-447. doi:10.1007/s00454-022-00412-w
apa: Fulek, R., & Kynčl, J. (2022). The Z2-Genus of Kuratowski minors. Discrete
and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00412-w
chicago: Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” Discrete
and Computational Geometry. Springer Nature, 2022. https://doi.org/10.1007/s00454-022-00412-w.
ieee: R. Fulek and J. Kynčl, “The Z2-Genus of Kuratowski minors,” Discrete and
Computational Geometry, vol. 68. Springer Nature, pp. 425–447, 2022.
ista: Fulek R, Kynčl J. 2022. The Z2-Genus of Kuratowski minors. Discrete and Computational
Geometry. 68, 425–447.
mla: Fulek, Radoslav, and Jan Kynčl. “The Z2-Genus of Kuratowski Minors.” Discrete
and Computational Geometry, vol. 68, Springer Nature, 2022, pp. 425–47, doi:10.1007/s00454-022-00412-w.
short: R. Fulek, J. Kynčl, Discrete and Computational Geometry 68 (2022) 425–447.
date_created: 2022-07-17T22:01:56Z
date_published: 2022-09-01T00:00:00Z
date_updated: 2025-04-14T13:52:37Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-022-00412-w
external_id:
arxiv:
- '1803.05085'
isi:
- '000825014500001'
intvolume: ' 68'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1803.05085
month: '09'
oa: 1
oa_version: Preprint
page: 425-447
project:
- _id: 261FA626-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M02281
name: Eliminating intersections in drawings of graphs
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '186'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: The Z2-Genus of Kuratowski minors
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 68
year: '2022'
...
---
_id: '12129'
abstract:
- lang: eng
text: 'Given a finite point set P in general position in the plane, a full triangulation
of P is a maximal straight-line embedded plane graph on P. A partial triangulation
of P is a full triangulation of some subset P′ of P containing all extreme points
in P. A bistellar flip on a partial triangulation either flips an edge (called
edge flip), removes a non-extreme point of degree 3, or adds a point in P∖P′ as
vertex of degree 3. The bistellar flip graph has all partial triangulations as
vertices, and a pair of partial triangulations is adjacent if they can be obtained
from one another by a bistellar flip. The edge flip graph is defined with full
triangulations as vertices, and edge flips determining the adjacencies. Lawson
showed in the early seventies that these graphs are connected. The goal of this
paper is to investigate the structure of these graphs, with emphasis on their
vertex connectivity. For sets P of n points in the plane in general position,
we show that the edge flip graph is ⌈n/2−2⌉-vertex connected, and the bistellar
flip graph is (n−3)-vertex connected; both results are tight. The latter bound
matches the situation for the subfamily of regular triangulations (i.e., partial
triangulations obtained by lifting the points to 3-space and projecting back the
lower convex hull), where (n−3)-vertex connectivity has been known since the late
eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky
and Balinski’s Theorem. For the edge flip-graph, we additionally show that the
vertex connectivity is at least as large as (and hence equal to) the minimum degree
(i.e., the minimum number of flippable edges in any full triangulation), provided
that n is large enough. Our methods also yield several other results: (i) The
edge flip graph can be covered by graphs of polytopes of dimension ⌈n/2−2⌉ (products
of associahedra) and the bistellar flip graph can be covered by graphs of polytopes
of dimension n−3 (products of secondary polytopes). (ii) A partial triangulation
is regular, if it has distance n−3 in the Hasse diagram of the partial order of
partial subdivisions from the trivial subdivision. (iii) All partial triangulations
of a point set are regular iff the partial order of partial subdivisions has height
n−3. (iv) There are arbitrarily large sets P with non-regular partial triangulations
and such that every proper subset has only regular triangulations, i.e., there
are no small certificates for the existence of non-regular triangulations.'
acknowledgement: "This is a full and revised version of [38] (on partial triangulations)
in Proceedings of the 36th Annual International Symposium on Computational Geometry
(SoCG‘20) and of some of the results in [37] (on full triangulations) in Proceedings
of the 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA‘20).\r\nThis
research started at the 11th Gremo’s Workshop on Open Problems (GWOP), Alp Sellamatt,
Switzerland, June 24–28, 2013, motivated by a question posed by Filip Mori´c on
full triangulations. Research was supported by the Swiss National Science Foundation
within the collaborative DACH project Arrangements and Drawings as SNSF Project
200021E-171681, and by IST Austria and Berlin Free University during a sabbatical
stay of the second author. We thank Michael Joswig, Jesús De Loera, and Francisco
Santos for helpful discussions on the topics of this paper, and Daniel Bertschinger
and Valentin Stoppiello for carefully reading earlier versions and for many helpful
comments.\r\nOpen access funding provided by the Swiss Federal Institute of Technology
Zürich"
article_processing_charge: No
article_type: original
author:
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
- first_name: Emo
full_name: Welzl, Emo
last_name: Welzl
citation:
ama: Wagner U, Welzl E. Connectivity of triangulation flip graphs in the plane.
Discrete & Computational Geometry. 2022;68(4):1227-1284. doi:10.1007/s00454-022-00436-2
apa: Wagner, U., & Welzl, E. (2022). Connectivity of triangulation flip graphs
in the plane. Discrete & Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-022-00436-2
chicago: Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs
in the Plane.” Discrete & Computational Geometry. Springer Nature,
2022. https://doi.org/10.1007/s00454-022-00436-2.
ieee: U. Wagner and E. Welzl, “Connectivity of triangulation flip graphs in the
plane,” Discrete & Computational Geometry, vol. 68, no. 4. Springer
Nature, pp. 1227–1284, 2022.
ista: Wagner U, Welzl E. 2022. Connectivity of triangulation flip graphs in the
plane. Discrete & Computational Geometry. 68(4), 1227–1284.
mla: Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs in the
Plane.” Discrete & Computational Geometry, vol. 68, no. 4, Springer
Nature, 2022, pp. 1227–84, doi:10.1007/s00454-022-00436-2.
short: U. Wagner, E. Welzl, Discrete & Computational Geometry 68 (2022) 1227–1284.
corr_author: '1'
date_created: 2023-01-12T12:02:28Z
date_published: 2022-11-14T00:00:00Z
date_updated: 2025-07-10T11:54:56Z
day: '14'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/s00454-022-00436-2
external_id:
isi:
- '000883222200003'
file:
- access_level: open_access
checksum: 307e879d09e52eddf5b225d0aaa9213a
content_type: application/pdf
creator: dernst
date_created: 2023-01-23T11:10:03Z
date_updated: 2023-01-23T11:10:03Z
file_id: '12345'
file_name: 2022_DiscreteCompGeometry_Wagner.pdf
file_size: 1747581
relation: main_file
success: 1
file_date_updated: 2023-01-23T11:10:03Z
has_accepted_license: '1'
intvolume: ' 68'
isi: 1
issue: '4'
keyword:
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Theoretical Computer Science
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
page: 1227-1284
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '7807'
relation: earlier_version
status: public
- id: '7990'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Connectivity of triangulation flip graphs in the plane
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 68
year: '2022'
...
---
_id: '10773'
abstract:
- lang: eng
text: The Voronoi tessellation in Rd is defined by locally minimizing the power
distance to given weighted points. Symmetrically, the Delaunay mosaic can be defined
by locally maximizing the negative power distance to other such points. We prove
that the average of the two piecewise quadratic functions is piecewise linear,
and that all three functions have the same critical points and values. Discretizing
the two piecewise quadratic functions, we get the alpha shapes as sublevel sets
of the discrete function on the Delaunay mosaic, and analogous shapes as superlevel
sets of the discrete function on the Voronoi tessellation. For the same non-critical
value, the corresponding shapes are disjoint, separated by a narrow channel that
contains no critical points but the entire level set of the piecewise linear function.
acknowledgement: Open access funding provided by the Institute of Science and Technology
(IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Ranita
full_name: Biswas, Ranita
id: 3C2B033E-F248-11E8-B48F-1D18A9856A87
last_name: Biswas
orcid: 0000-0002-5372-7890
- first_name: Sebastiano
full_name: Cultrera Di Montesano, Sebastiano
id: 34D2A09C-F248-11E8-B48F-1D18A9856A87
last_name: Cultrera Di Montesano
orcid: 0000-0001-6249-0832
- first_name: Herbert
full_name: Edelsbrunner, Herbert
id: 3FB178DA-F248-11E8-B48F-1D18A9856A87
last_name: Edelsbrunner
orcid: 0000-0002-9823-6833
- first_name: Morteza
full_name: Saghafian, Morteza
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Continuous
and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete
and Computational Geometry. 2022;67:811-842. doi:10.1007/s00454-022-00371-2
apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M.
(2022). Continuous and discrete radius functions on Voronoi tessellations and
Delaunay mosaics. Discrete and Computational Geometry. Springer Nature.
https://doi.org/10.1007/s00454-022-00371-2
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Continuous and Discrete Radius Functions on Voronoi Tessellations
and Delaunay Mosaics.” Discrete and Computational Geometry. Springer Nature,
2022. https://doi.org/10.1007/s00454-022-00371-2.
ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Continuous
and discrete radius functions on Voronoi tessellations and Delaunay mosaics,”
Discrete and Computational Geometry, vol. 67. Springer Nature, pp. 811–842,
2022.
ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2022. Continuous
and discrete radius functions on Voronoi tessellations and Delaunay mosaics. Discrete
and Computational Geometry. 67, 811–842.
mla: Biswas, Ranita, et al. “Continuous and Discrete Radius Functions on Voronoi
Tessellations and Delaunay Mosaics.” Discrete and Computational Geometry,
vol. 67, Springer Nature, 2022, pp. 811–42, doi:10.1007/s00454-022-00371-2.
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Discrete
and Computational Geometry 67 (2022) 811–842.
corr_author: '1'
date_created: 2022-02-20T23:01:34Z
date_published: 2022-04-01T00:00:00Z
date_updated: 2024-10-09T21:01:38Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-022-00371-2
external_id:
isi:
- '000752175300002'
file:
- access_level: open_access
checksum: 9383d3b70561bacee905e335dc922680
content_type: application/pdf
creator: dernst
date_created: 2022-08-02T06:07:55Z
date_updated: 2022-08-02T06:07:55Z
file_id: '11718'
file_name: 2022_DiscreteCompGeometry_Biswas.pdf
file_size: 2518111
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file_date_updated: 2022-08-02T06:07:55Z
has_accepted_license: '1'
intvolume: ' 67'
isi: 1
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
page: 811-842
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Continuous and discrete radius functions on Voronoi tessellations and Delaunay
mosaics
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 67
year: '2022'
...
---
_id: '10776'
abstract:
- lang: eng
text: 'Let K be a convex body in Rn (i.e., a compact convex set with nonempty interior).
Given a point p in the interior of K, a hyperplane h passing through p is called
barycentric if p is the barycenter of K∩h. In 1961, Grünbaum raised the question
whether, for every K, there exists an interior point p through which there are
at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly
resolved affirmatively by showing that this is the case if p=p0 is the point of
maximal depth in K. However, while working on a related question, we noticed that
one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample;
this re-opens Grünbaum’s question. It follows from known results that for n≥2,
there are always at least three distinct barycentric cuts through the point p0∈K
of maximal depth. Using tools related to Morse theory we are able to improve this
bound: four distinct barycentric cuts through p0 are guaranteed if n≥3.'
acknowledgement: The work by Zuzana Patáková has been partially supported by Charles
University Research Center Program No. UNCE/SCI/022, and part of it was done during
her research stay at IST Austria. The work by Martin Tancer is supported by the
GAČR Grant 19-04113Y and by the Charles University Projects PRIMUS/17/SCI/3 and
UNCE/SCI/004.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Zuzana
full_name: Patakova, Zuzana
id: 48B57058-F248-11E8-B48F-1D18A9856A87
last_name: Patakova
orcid: 0000-0002-3975-1683
- first_name: Martin
full_name: Tancer, Martin
last_name: Tancer
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Patakova Z, Tancer M, Wagner U. Barycentric cuts through a convex body. Discrete
and Computational Geometry. 2022;68:1133-1154. doi:10.1007/s00454-021-00364-7
apa: Patakova, Z., Tancer, M., & Wagner, U. (2022). Barycentric cuts through
a convex body. Discrete and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-021-00364-7
chicago: Patakova, Zuzana, Martin Tancer, and Uli Wagner. “Barycentric Cuts through
a Convex Body.” Discrete and Computational Geometry. Springer Nature, 2022.
https://doi.org/10.1007/s00454-021-00364-7.
ieee: Z. Patakova, M. Tancer, and U. Wagner, “Barycentric cuts through a convex
body,” Discrete and Computational Geometry, vol. 68. Springer Nature, pp.
1133–1154, 2022.
ista: Patakova Z, Tancer M, Wagner U. 2022. Barycentric cuts through a convex body.
Discrete and Computational Geometry. 68, 1133–1154.
mla: Patakova, Zuzana, et al. “Barycentric Cuts through a Convex Body.” Discrete
and Computational Geometry, vol. 68, Springer Nature, 2022, pp. 1133–54, doi:10.1007/s00454-021-00364-7.
short: Z. Patakova, M. Tancer, U. Wagner, Discrete and Computational Geometry 68
(2022) 1133–1154.
date_created: 2022-02-20T23:01:35Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-02T14:38:58Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-021-00364-7
external_id:
arxiv:
- '2003.13536'
isi:
- '000750681500001'
intvolume: ' 68'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2003.13536
month: '12'
oa: 1
oa_version: Preprint
page: 1133-1154
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Barycentric cuts through a convex body
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 68
year: '2022'
...
---
_id: '11446'
abstract:
- lang: eng
text: Suppose that n is not a prime power and not twice a prime power. We prove
that for any Hausdorff compactum X with a free action of the symmetric group Sn,
there exists an Sn-equivariant map X→Rn whose image avoids the diagonal {(x,x,…,x)∈Rn∣x∈R}.
Previously, the special cases of this statement for certain X were usually proved
using the equivartiant obstruction theory. Such calculations are difficult and
may become infeasible past the first (primary) obstruction. We take a different
approach which allows us to prove the vanishing of all obstructions simultaneously.
The essential step in the proof is classifying the possible degrees of Sn-equivariant
maps from the boundary ∂Δn−1 of (n−1)-simplex to itself. Existence of equivariant
maps between spaces is important for many questions arising from discrete mathematics
and geometry, such as Kneser’s conjecture, the Square Peg conjecture, the Splitting
Necklace problem, and the Topological Tverberg conjecture, etc. We demonstrate
the utility of our result applying it to one such question, a specific instance
of envy-free division problem.
acknowledgement: S. Avvakumov has received funding from the European Research Council
under the European Union’s Seventh Framework Programme ERC Grant agreement ERC StG
716424–CASe. S. Kudrya was supported by the Austrian Academic Exchange Service (OeAD),
ICM-2019-13577.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
orcid: 0000-0002-7840-5062
- first_name: Sergey
full_name: Kudrya, Sergey
id: ecf01965-d252-11ea-95a5-8ada5f6c6a67
last_name: Kudrya
citation:
ama: Avvakumov S, Kudrya S. Vanishing of all equivariant obstructions and the mapping
degree. Discrete & Computational Geometry. 2021;66(3):1202-1216. doi:10.1007/s00454-021-00299-z
apa: Avvakumov, S., & Kudrya, S. (2021). Vanishing of all equivariant obstructions
and the mapping degree. Discrete & Computational Geometry. Springer
Nature. https://doi.org/10.1007/s00454-021-00299-z
chicago: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
and the Mapping Degree.” Discrete & Computational Geometry. Springer
Nature, 2021. https://doi.org/10.1007/s00454-021-00299-z.
ieee: S. Avvakumov and S. Kudrya, “Vanishing of all equivariant obstructions and
the mapping degree,” Discrete & Computational Geometry, vol. 66, no.
3. Springer Nature, pp. 1202–1216, 2021.
ista: Avvakumov S, Kudrya S. 2021. Vanishing of all equivariant obstructions and
the mapping degree. Discrete & Computational Geometry. 66(3), 1202–1216.
mla: Avvakumov, Sergey, and Sergey Kudrya. “Vanishing of All Equivariant Obstructions
and the Mapping Degree.” Discrete & Computational Geometry, vol. 66,
no. 3, Springer Nature, 2021, pp. 1202–16, doi:10.1007/s00454-021-00299-z.
short: S. Avvakumov, S. Kudrya, Discrete & Computational Geometry 66 (2021)
1202–1216.
corr_author: '1'
date_created: 2022-06-17T08:45:15Z
date_published: 2021-10-01T00:00:00Z
date_updated: 2025-04-14T09:10:06Z
day: '01'
doi: 10.1007/s00454-021-00299-z
extern: '1'
external_id:
arxiv:
- '1910.12628'
intvolume: ' 66'
issue: '3'
keyword:
- Computational Theory and Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Theoretical Computer Science
language:
- iso: eng
month: '10'
oa_version: Preprint
page: 1202-1216
publication: Discrete & Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '8182'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Vanishing of all equivariant obstructions and the mapping degree
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 66
year: '2021'
...
---
_id: '7905'
abstract:
- lang: eng
text: We investigate a sheaf-theoretic interpretation of stratification learning
from geometric and topological perspectives. Our main result is the construction
of stratification learning algorithms framed in terms of a sheaf on a partially
ordered set with the Alexandroff topology. We prove that the resulting decomposition
is the unique minimal stratification for which the strata are homogeneous and
the given sheaf is constructible. In particular, when we choose to work with the
local homology sheaf, our algorithm gives an alternative to the local homology
transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM
Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the
cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2),
195–222, 2020). Additionally, we give examples of stratifications based on the
geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018),
illustrating how the sheaf-theoretic approach can be used to study stratifications
from both topological and geometric perspectives. This approach also points toward
future applications of sheaf theory in the study of topological data analysis
by illustrating the utility of the language of sheaf theory in generalizing existing
algorithms.
acknowledgement: Open access funding provided by Institute of Science and Technology
(IST Austria). This work was partially supported by NSF IIS-1513616 and NSF ABI-1661375.
The authors would like to thank the anonymous referees for their insightful comments.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Adam
full_name: Brown, Adam
id: 70B7FDF6-608D-11E9-9333-8535E6697425
last_name: Brown
- first_name: Bei
full_name: Wang, Bei
last_name: Wang
citation:
ama: Brown A, Wang B. Sheaf-theoretic stratification learning from geometric and
topological perspectives. Discrete and Computational Geometry. 2021;65:1166-1198.
doi:10.1007/s00454-020-00206-y
apa: Brown, A., & Wang, B. (2021). Sheaf-theoretic stratification learning from
geometric and topological perspectives. Discrete and Computational Geometry.
Springer Nature. https://doi.org/10.1007/s00454-020-00206-y
chicago: Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from
Geometric and Topological Perspectives.” Discrete and Computational Geometry.
Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00206-y.
ieee: A. Brown and B. Wang, “Sheaf-theoretic stratification learning from geometric
and topological perspectives,” Discrete and Computational Geometry, vol.
65. Springer Nature, pp. 1166–1198, 2021.
ista: Brown A, Wang B. 2021. Sheaf-theoretic stratification learning from geometric
and topological perspectives. Discrete and Computational Geometry. 65, 1166–1198.
mla: Brown, Adam, and Bei Wang. “Sheaf-Theoretic Stratification Learning from Geometric
and Topological Perspectives.” Discrete and Computational Geometry, vol.
65, Springer Nature, 2021, pp. 1166–98, doi:10.1007/s00454-020-00206-y.
short: A. Brown, B. Wang, Discrete and Computational Geometry 65 (2021) 1166–1198.
corr_author: '1'
date_created: 2020-05-30T10:26:04Z
date_published: 2021-06-01T00:00:00Z
date_updated: 2025-04-15T06:53:15Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-020-00206-y
external_id:
arxiv:
- '1712.07734'
isi:
- '000536324700001'
file:
- access_level: open_access
checksum: 487a84ea5841b75f04f66d7ebd71b67e
content_type: application/pdf
creator: dernst
date_created: 2020-11-25T09:06:41Z
date_updated: 2020-11-25T09:06:41Z
file_id: '8803'
file_name: 2020_DiscreteCompGeometry_Brown.pdf
file_size: 1013730
relation: main_file
success: 1
file_date_updated: 2020-11-25T09:06:41Z
has_accepted_license: '1'
intvolume: ' 65'
isi: 1
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 1166-1198
project:
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sheaf-theoretic stratification learning from geometric and topological perspectives
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 65
year: '2021'
...
---
_id: '8248'
abstract:
- lang: eng
text: 'We consider the following setting: suppose that we are given a manifold M
in Rd with positive reach. Moreover assume that we have an embedded simplical
complex A without boundary, whose vertex set lies on the manifold, is sufficiently
dense and such that all simplices in A have sufficient quality. We prove that
if, locally, interiors of the projection of the simplices onto the tangent space
do not intersect, then A is a triangulation of the manifold, that is, they are
homeomorphic.'
acknowledgement: "Open access funding provided by the Institute of Science and Technology
(IST Austria). Arijit Ghosh is supported by the Ramanujan Fellowship (No. SB/S2/RJN-064/2015),
India.\r\nThis work has been funded by the European Research Council under the European
Union’s ERC Grant Agreement number 339025 GUDHI (Algorithmic Foundations of Geometric
Understanding in Higher Dimensions). The third author is supported by Ramanujan
Fellowship (No. SB/S2/RJN-064/2015), India. The fifth author also received funding
from the European Union’s Horizon 2020 research and innovation programme under the
Marie Skłodowska-Curie Grant Agreement No. 754411."
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- first_name: Ramsay
full_name: Dyer, Ramsay
last_name: Dyer
- first_name: Arijit
full_name: Ghosh, Arijit
last_name: Ghosh
- first_name: Andre
full_name: Lieutier, Andre
last_name: Lieutier
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. Local conditions
for triangulating submanifolds of Euclidean space. Discrete and Computational
Geometry. 2021;66:666-686. doi:10.1007/s00454-020-00233-9
apa: Boissonnat, J.-D., Dyer, R., Ghosh, A., Lieutier, A., & Wintraecken, M.
(2021). Local conditions for triangulating submanifolds of Euclidean space. Discrete
and Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00233-9
chicago: Boissonnat, Jean-Daniel, Ramsay Dyer, Arijit Ghosh, Andre Lieutier, and
Mathijs Wintraecken. “Local Conditions for Triangulating Submanifolds of Euclidean
Space.” Discrete and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00233-9.
ieee: J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, and M. Wintraecken, “Local
conditions for triangulating submanifolds of Euclidean space,” Discrete and
Computational Geometry, vol. 66. Springer Nature, pp. 666–686, 2021.
ista: Boissonnat J-D, Dyer R, Ghosh A, Lieutier A, Wintraecken M. 2021. Local conditions
for triangulating submanifolds of Euclidean space. Discrete and Computational
Geometry. 66, 666–686.
mla: Boissonnat, Jean-Daniel, et al. “Local Conditions for Triangulating Submanifolds
of Euclidean Space.” Discrete and Computational Geometry, vol. 66, Springer
Nature, 2021, pp. 666–86, doi:10.1007/s00454-020-00233-9.
short: J.-D. Boissonnat, R. Dyer, A. Ghosh, A. Lieutier, M. Wintraecken, Discrete
and Computational Geometry 66 (2021) 666–686.
corr_author: '1'
date_created: 2020-08-11T07:11:51Z
date_published: 2021-09-01T00:00:00Z
date_updated: 2025-04-14T07:44:05Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1007/s00454-020-00233-9
ec_funded: 1
external_id:
isi:
- '000558119300001'
has_accepted_license: '1'
intvolume: ' 66'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s00454-020-00233-9
month: '09'
oa: 1
oa_version: Published Version
page: 666-686
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Local conditions for triangulating submanifolds of Euclidean space
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 66
year: '2021'
...
---
_id: '8338'
abstract:
- lang: eng
text: Canonical parametrisations of classical confocal coordinate systems are introduced
and exploited to construct non-planar analogues of incircular (IC) nets on individual
quadrics and systems of confocal quadrics. Intimate connections with classical
deformations of quadrics that are isometric along asymptotic lines and circular
cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces
of Blaschke type generated by asymptotic and characteristic lines that are diagonally
related to lines of curvature is proved theoretically and established constructively.
Appropriate samplings (grids) of these webs lead to three-dimensional extensions
of non-planar IC nets. Three-dimensional octahedral grids composed of planes and
spatially extending (checkerboard) IC-nets are shown to arise in connection with
systems of confocal quadrics in Minkowski space. In this context, the Laguerre
geometric notion of conical octahedral grids of planes is introduced. The latter
generalise the octahedral grids derived from systems of confocal quadrics in Minkowski
space. An explicit construction of conical octahedral grids is presented. The
results are accompanied by various illustrations which are based on the explicit
formulae provided by the theory.
acknowledgement: This research was supported by the DFG Collaborative Research Center
TRR 109 “Discretization in Geometry and Dynamics”. W.K.S. was also supported by
the Australian Research Council (DP1401000851). A.V.A. was also supported by the
European Research Council (ERC) under the European Union’s Horizon 2020 research
and innovation programme (Grant Agreement No. 78818 Alpha).
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Arseniy
full_name: Akopyan, Arseniy
id: 430D2C90-F248-11E8-B48F-1D18A9856A87
last_name: Akopyan
orcid: 0000-0002-2548-617X
- first_name: Alexander I.
full_name: Bobenko, Alexander I.
last_name: Bobenko
- first_name: Wolfgang K.
full_name: Schief, Wolfgang K.
last_name: Schief
- first_name: Jan
full_name: Techter, Jan
last_name: Techter
citation:
ama: Akopyan A, Bobenko AI, Schief WK, Techter J. On mutually diagonal nets on (confocal)
quadrics and 3-dimensional webs. Discrete and Computational Geometry. 2021;66:938-976.
doi:10.1007/s00454-020-00240-w
apa: Akopyan, A., Bobenko, A. I., Schief, W. K., & Techter, J. (2021). On mutually
diagonal nets on (confocal) quadrics and 3-dimensional webs. Discrete and Computational
Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00240-w
chicago: Akopyan, Arseniy, Alexander I. Bobenko, Wolfgang K. Schief, and Jan Techter.
“On Mutually Diagonal Nets on (Confocal) Quadrics and 3-Dimensional Webs.” Discrete
and Computational Geometry. Springer Nature, 2021. https://doi.org/10.1007/s00454-020-00240-w.
ieee: A. Akopyan, A. I. Bobenko, W. K. Schief, and J. Techter, “On mutually diagonal
nets on (confocal) quadrics and 3-dimensional webs,” Discrete and Computational
Geometry, vol. 66. Springer Nature, pp. 938–976, 2021.
ista: Akopyan A, Bobenko AI, Schief WK, Techter J. 2021. On mutually diagonal nets
on (confocal) quadrics and 3-dimensional webs. Discrete and Computational Geometry.
66, 938–976.
mla: Akopyan, Arseniy, et al. “On Mutually Diagonal Nets on (Confocal) Quadrics
and 3-Dimensional Webs.” Discrete and Computational Geometry, vol. 66,
Springer Nature, 2021, pp. 938–76, doi:10.1007/s00454-020-00240-w.
short: A. Akopyan, A.I. Bobenko, W.K. Schief, J. Techter, Discrete and Computational
Geometry 66 (2021) 938–976.
date_created: 2020-09-06T22:01:13Z
date_published: 2021-10-01T00:00:00Z
date_updated: 2025-04-14T07:48:36Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/s00454-020-00240-w
ec_funded: 1
external_id:
arxiv:
- '1908.00856'
isi:
- '000564488500002'
intvolume: ' 66'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1908.00856
month: '10'
oa: 1
oa_version: Preprint
page: 938-976
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 66
year: '2021'
...
---
_id: '8940'
abstract:
- lang: eng
text: We quantise Whitney’s construction to prove the existence of a triangulation
for any C^2 manifold, so that we get an algorithm with explicit bounds. We also
give a new elementary proof, which is completely geometric.
acknowledgement: This work has been funded by the European Research Council under
the European Union’s ERC Grant Agreement Number 339025 GUDHI (Algorithmic Foundations
of Geometric Understanding in Higher Dimensions). The third author also received
funding from the European Union’s Horizon 2020 research and innovation programme
under the Marie Skłodowska-Curie Grant Agreement No. 754411. Open access funding
provided by the Institute of Science and Technology (IST Austria).
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Jean-Daniel
full_name: Boissonnat, Jean-Daniel
last_name: Boissonnat
- first_name: Siargey
full_name: Kachanovich, Siargey
last_name: Kachanovich
- first_name: Mathijs
full_name: Wintraecken, Mathijs
id: 307CFBC8-F248-11E8-B48F-1D18A9856A87
last_name: Wintraecken
orcid: 0000-0002-7472-2220
citation:
ama: 'Boissonnat J-D, Kachanovich S, Wintraecken M. Triangulating submanifolds:
An elementary and quantified version of Whitney’s method. Discrete & Computational
Geometry. 2021;66(1):386-434. doi:10.1007/s00454-020-00250-8'
apa: 'Boissonnat, J.-D., Kachanovich, S., & Wintraecken, M. (2021). Triangulating
submanifolds: An elementary and quantified version of Whitney’s method. Discrete
& Computational Geometry. Springer Nature. https://doi.org/10.1007/s00454-020-00250-8'
chicago: 'Boissonnat, Jean-Daniel, Siargey Kachanovich, and Mathijs Wintraecken.
“Triangulating Submanifolds: An Elementary and Quantified Version of Whitney’s
Method.” Discrete & Computational Geometry. Springer Nature, 2021.
https://doi.org/10.1007/s00454-020-00250-8.'
ieee: 'J.-D. Boissonnat, S. Kachanovich, and M. Wintraecken, “Triangulating submanifolds:
An elementary and quantified version of Whitney’s method,” Discrete & Computational
Geometry, vol. 66, no. 1. Springer Nature, pp. 386–434, 2021.'
ista: 'Boissonnat J-D, Kachanovich S, Wintraecken M. 2021. Triangulating submanifolds:
An elementary and quantified version of Whitney’s method. Discrete & Computational
Geometry. 66(1), 386–434.'
mla: 'Boissonnat, Jean-Daniel, et al. “Triangulating Submanifolds: An Elementary
and Quantified Version of Whitney’s Method.” Discrete & Computational Geometry,
vol. 66, no. 1, Springer Nature, 2021, pp. 386–434, doi:10.1007/s00454-020-00250-8.'
short: J.-D. Boissonnat, S. Kachanovich, M. Wintraecken, Discrete & Computational
Geometry 66 (2021) 386–434.
corr_author: '1'
date_created: 2020-12-12T11:07:02Z
date_published: 2021-07-01T00:00:00Z
date_updated: 2025-04-14T07:43:50Z
day: '01'
ddc:
- '516'
department:
- _id: HeEd
doi: 10.1007/s00454-020-00250-8
ec_funded: 1
external_id:
isi:
- '000597770300001'
file:
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date_created: 2021-08-06T09:52:29Z
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keyword:
- Theoretical Computer Science
- Computational Theory and Mathematics
- Geometry and Topology
- Discrete Mathematics and Combinatorics
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 386-434
project:
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call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
publication: Discrete & Computational Geometry
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- 1432-0444
issn:
- 0179-5376
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
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title: 'Triangulating submanifolds: An elementary and quantified version of Whitney’s
method'
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legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 66
year: '2021'
...