---
_id: '15012'
abstract:
- lang: eng
text: We solve a problem of Dujmović and Wood (2007) by showing that a complete
convex geometric graph on n vertices cannot be decomposed into fewer than n-1
star-forests, each consisting of noncrossing edges. This bound is clearly tight.
We also discuss similar questions for abstract graphs.
acknowledgement: János Pach’s Research partially supported by European Research Council
(ERC), grant “GeoScape” No. 882971 and by the Hungarian Science Foundation (NKFIH),
grant K-131529. Work by Morteza Saghafian is partially supported by the European
Research Council (ERC), grant No. 788183, and by the Wittgenstein Prize, Austrian
Science Fund (FWF), grant No. Z 342-N31.
alternative_title:
- LNCS
article_processing_charge: No
arxiv: 1
author:
- first_name: János
full_name: Pach, János
id: E62E3130-B088-11EA-B919-BF823C25FEA4
last_name: Pach
- first_name: Morteza
full_name: Saghafian, Morteza
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
- first_name: Patrick
full_name: Schnider, Patrick
last_name: Schnider
citation:
ama: 'Pach J, Saghafian M, Schnider P. Decomposition of geometric graphs into star-forests.
In: 31st International Symposium on Graph Drawing and Network Visualization.
Vol 14465. Springer Nature; 2024:339-346. doi:10.1007/978-3-031-49272-3_23'
apa: 'Pach, J., Saghafian, M., & Schnider, P. (2024). Decomposition of geometric
graphs into star-forests. In 31st International Symposium on Graph Drawing
and Network Visualization (Vol. 14465, pp. 339–346). Isola delle Femmine,
Palermo, Italy: Springer Nature. https://doi.org/10.1007/978-3-031-49272-3_23'
chicago: Pach, János, Morteza Saghafian, and Patrick Schnider. “Decomposition of Geometric
Graphs into Star-Forests.” In 31st International Symposium on Graph Drawing
and Network Visualization, 14465:339–46. Springer Nature, 2024. https://doi.org/10.1007/978-3-031-49272-3_23.
ieee: J. Pach, M. Saghafian, and P. Schnider, “Decomposition of geometric graphs
into star-forests,” in 31st International Symposium on Graph Drawing and Network
Visualization, Isola delle Femmine, Palermo, Italy, 2024, vol. 14465, pp.
339–346.
ista: 'Pach J, Saghafian M, Schnider P. 2024. Decomposition of geometric graphs
into star-forests. 31st International Symposium on Graph Drawing and Network Visualization.
GD: Graph Drawing and Network Visualization, LNCS, vol. 14465, 339–346.'
mla: Pach, János, et al. “Decomposition of Geometric Graphs into Star-Forests.”
31st International Symposium on Graph Drawing and Network Visualization,
vol. 14465, Springer Nature, 2024, pp. 339–46, doi:10.1007/978-3-031-49272-3_23.
short: J. Pach, M. Saghafian, P. Schnider, in:, 31st International Symposium on
Graph Drawing and Network Visualization, Springer Nature, 2024, pp. 339–346.
conference:
end_date: 2023-09-22
location: Isola delle Femmine, Palermo, Italy
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2023-09-20
date_created: 2024-02-18T23:01:03Z
date_published: 2024-01-01T00:00:00Z
date_updated: 2024-02-20T09:13:07Z
day: '01'
department:
- _id: HeEd
doi: 10.1007/978-3-031-49272-3_23
ec_funded: 1
external_id:
arxiv:
- '2306.13201'
intvolume: ' 14465'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2306.13201
month: '01'
oa: 1
oa_version: Preprint
page: 339-346
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: The Wittgenstein Prize
publication: 31st International Symposium on Graph Drawing and Network Visualization
publication_identifier:
eissn:
- '16113349'
isbn:
- '9783031492716'
issn:
- '03029743'
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Decomposition of geometric graphs into star-forests
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 14465
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. 2023. doi:10.1007/s00454-023-00566-1
apa: Edelsbrunner, H., Garber, A., Ghafari, M., Heiss, T., & Saghafian, M. (2023).
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,
2023. 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. Springer Nature, 2023.
ista: Edelsbrunner H, Garber A, Ghafari M, Heiss T, Saghafian M. 2023. On angles
in higher order Brillouin tessellations and related tilings in the plane. Discrete
and Computational Geometry.
mla: Edelsbrunner, Herbert, et al. “On Angles in Higher Order Brillouin Tessellations
and Related Tilings in the Plane.” Discrete and Computational Geometry,
Springer Nature, 2023, doi:10.1007/s00454-023-00566-1.
short: H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Discrete
and Computational Geometry (2023).
date_created: 2023-09-17T22:01:10Z
date_published: 2023-09-07T00:00:00Z
date_updated: 2023-12-13T12:25:06Z
day: '07'
department:
- _id: HeEd
doi: 10.1007/s00454-023-00566-1
ec_funded: 1
external_id:
arxiv:
- '2204.01076'
isi:
- '001060727600004'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.1007/s00454-023-00566-1
month: '09'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: The Wittgenstein Prize
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Discrete and Computational Geometry
publication_identifier:
eissn:
- 1432-0444
issn:
- 0179-5376
publication_status: epub_ahead
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
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '13182'
abstract:
- lang: eng
text: "We characterize critical points of 1-dimensional maps paired in persistent
homology\r\ngeometrically and this way get elementary proofs of theorems about
the symmetry\r\nof persistence diagrams and the variation of such maps. In particular,
we identify\r\nbranching points and endpoints of networks as the sole source of
asymmetry and\r\nrelate the cycle basis in persistent homology with a version
of the stable marriage\r\nproblem. Our analysis provides the foundations of fast
algorithms for maintaining a\r\ncollection of sorted lists together with its persistence
diagram."
acknowledgement: Open access funding provided by Austrian Science Fund (FWF). 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. The authors of
this paper thank anonymous reviewers for their constructive criticism and Monika
Henzinger for detailed comments on an earlier version of this paper.
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
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Geometric characterization
of the persistence of 1D maps. Journal of Applied and Computational Topology.
2023. doi:10.1007/s41468-023-00126-9
apa: Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian, M.
(2023). Geometric characterization of the persistence of 1D maps. Journal of
Applied and Computational Topology. Springer Nature. https://doi.org/10.1007/s41468-023-00126-9
chicago: Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Geometric Characterization of the Persistence of 1D Maps.”
Journal of Applied and Computational Topology. Springer Nature, 2023. https://doi.org/10.1007/s41468-023-00126-9.
ieee: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Geometric
characterization of the persistence of 1D maps,” Journal of Applied and Computational
Topology. Springer Nature, 2023.
ista: Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. 2023. Geometric
characterization of the persistence of 1D maps. Journal of Applied and Computational
Topology.
mla: Biswas, Ranita, et al. “Geometric Characterization of the Persistence of 1D
Maps.” Journal of Applied and Computational Topology, Springer Nature,
2023, doi:10.1007/s41468-023-00126-9.
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Journal
of Applied and Computational Topology (2023).
date_created: 2023-07-02T22:00:44Z
date_published: 2023-06-17T00:00:00Z
date_updated: 2023-10-18T08:13:10Z
day: '17'
ddc:
- '000'
department:
- _id: HeEd
doi: 10.1007/s41468-023-00126-9
ec_funded: 1
file:
- access_level: open_access
checksum: 697249d5d1c61dea4410b9f021b70fce
content_type: application/pdf
creator: alisjak
date_created: 2023-07-03T09:41:05Z
date_updated: 2023-07-03T09:41:05Z
file_id: '13185'
file_name: 2023_Journal of Applied and Computational Topology_Biswas.pdf
file_size: 487355
relation: main_file
success: 1
file_date_updated: 2023-07-03T09:41:05Z
has_accepted_license: '1'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 0aa4bc98-070f-11eb-9043-e6fff9c6a316
grant_number: I4887
name: Discretization in Geometry and Dynamics
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: The Wittgenstein Prize
publication: Journal of Applied and Computational Topology
publication_identifier:
eissn:
- 2367-1734
issn:
- 2367-1726
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Geometric characterization of the persistence of 1D maps
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
year: '2023'
...
---
_id: '11658'
abstract:
- lang: eng
text: The depth of a cell in an arrangement of n (non-vertical) great-spheres in
Sd is the number of great-spheres that pass above the cell. We prove Euler-type
relations, which imply extensions of the classic Dehn–Sommerville relations for
convex polytopes to sublevel sets of the depth function, and we use the relations
to extend the expressions for the number of faces of neighborly polytopes to the
number of cells of levels in neighborly arrangements.
acknowledgement: 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: No
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
id: f86f7148-b140-11ec-9577-95435b8df824
last_name: Saghafian
citation:
ama: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in arrangements:
Dehn–Sommerville–Euler relations with applications. Leibniz International Proceedings
on Mathematics.'
apa: 'Biswas, R., Cultrera di Montesano, S., Edelsbrunner, H., & Saghafian,
M. (n.d.). Depth in arrangements: Dehn–Sommerville–Euler relations with applications.
Leibniz International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz
Zentrum für Informatik.'
chicago: 'Biswas, Ranita, Sebastiano Cultrera di Montesano, Herbert Edelsbrunner,
and Morteza Saghafian. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
with Applications.” Leibniz International Proceedings on Mathematics. Schloss
Dagstuhl - Leibniz Zentrum für Informatik, n.d.'
ieee: 'R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, and M. Saghafian, “Depth
in arrangements: Dehn–Sommerville–Euler relations with applications,” Leibniz
International Proceedings on Mathematics. Schloss Dagstuhl - Leibniz Zentrum
für Informatik.'
ista: 'Biswas R, Cultrera di Montesano S, Edelsbrunner H, Saghafian M. Depth in
arrangements: Dehn–Sommerville–Euler relations with applications. Leibniz International
Proceedings on Mathematics.'
mla: 'Biswas, Ranita, et al. “Depth in Arrangements: Dehn–Sommerville–Euler Relations
with Applications.” Leibniz International Proceedings on Mathematics, Schloss
Dagstuhl - Leibniz Zentrum für Informatik.'
short: R. Biswas, S. Cultrera di Montesano, H. Edelsbrunner, M. Saghafian, Leibniz
International Proceedings on Mathematics (n.d.).
date_created: 2022-07-27T09:27:34Z
date_published: 2022-07-27T00:00:00Z
date_updated: 2022-07-28T07:57:48Z
day: '27'
ddc:
- '510'
department:
- _id: GradSch
- _id: HeEd
ec_funded: 1
file:
- access_level: open_access
checksum: b2f511e8b1cae5f1892b0cdec341acac
content_type: application/pdf
creator: scultrer
date_created: 2022-07-27T09:25:53Z
date_updated: 2022-07-27T09:25:53Z
file_id: '11659'
file_name: D-S-E.pdf
file_size: 639266
relation: main_file
file_date_updated: 2022-07-27T09:25:53Z
has_accepted_license: '1'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Submitted Version
project:
- _id: 266A2E9E-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '788183'
name: Alpha Shape Theory Extended
- _id: 268116B8-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z00342
name: The Wittgenstein Prize
- _id: 2561EBF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: I02979-N35
name: Persistence and stability of geometric complexes
publication: Leibniz International Proceedings on Mathematics
publication_status: submitted
publisher: Schloss Dagstuhl - Leibniz Zentrum für Informatik
quality_controlled: '1'
status: public
title: 'Depth in arrangements: Dehn–Sommerville–Euler relations with applications'
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
year: '2022'
...