---
_id: '1381'
abstract:
- lang: eng
text: 'Motivated by Tverberg-type problems in topological combinatorics and by classical
results about embeddings (maps without double points), we study the question whether
a finite simplicial complex K can be mapped into double-struck Rd without higher-multiplicity
intersections. We focus on conditions for the existence of almost r-embeddings,
i.e., maps f : K → double-struck Rd such that f(σ1) ∩ ⋯ ∩ f(σr) = ∅ whenever σ1,
..., σr are pairwise disjoint simplices of K. Generalizing the classical Haefliger-Weber
embeddability criterion, we show that a well-known necessary deleted product condition
for the existence of almost r-embeddings is sufficient in a suitable r-metastable
range of dimensions: If rd ≥ (r + 1) dim K + 3, then there exists an almost r-embedding
K → double-struck Rd if and only if there exists an equivariant map (K)Δ r → Sr
Sd(r-1)-1, where (K)Δ r is the deleted r-fold product of K, the target Sd(r-1)-1
is the sphere of dimension d(r - 1) - 1, and Sr is the symmetric group. This significantly
extends one of the main results of our previous paper (which treated the special
case where d = rk and dim K = (r - 1)k for some k ≥ 3), and settles an open question
raised there.'
alternative_title:
- LIPIcs
author:
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Mabillard I, Wagner U. Eliminating higher-multiplicity intersections, II.
The deleted product criterion in the r-metastable range. In: Vol 51. Schloss Dagstuhl-
Leibniz-Zentrum fur Informatik GmbH; 2016:51.1-51.12. doi:10.4230/LIPIcs.SoCG.2016.51'
apa: 'Mabillard, I., & Wagner, U. (2016). Eliminating higher-multiplicity intersections,
II. The deleted product criterion in the r-metastable range (Vol. 51, p. 51.1-51.12).
Presented at the SoCG: Symposium on Computational Geometry, Medford, MA, USA:
Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH. https://doi.org/10.4230/LIPIcs.SoCG.2016.51'
chicago: Mabillard, Isaac, and Uli Wagner. “Eliminating Higher-Multiplicity Intersections,
II. The Deleted Product Criterion in the r-Metastable Range,” 51:51.1-51.12. Schloss
Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016. https://doi.org/10.4230/LIPIcs.SoCG.2016.51.
ieee: 'I. Mabillard and U. Wagner, “Eliminating higher-multiplicity intersections,
II. The deleted product criterion in the r-metastable range,” presented at the
SoCG: Symposium on Computational Geometry, Medford, MA, USA, 2016, vol. 51, p.
51.1-51.12.'
ista: 'Mabillard I, Wagner U. 2016. Eliminating higher-multiplicity intersections,
II. The deleted product criterion in the r-metastable range. SoCG: Symposium on
Computational Geometry, LIPIcs, vol. 51, 51.1-51.12.'
mla: Mabillard, Isaac, and Uli Wagner. Eliminating Higher-Multiplicity Intersections,
II. The Deleted Product Criterion in the r-Metastable Range. Vol. 51, Schloss
Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, 2016, p. 51.1-51.12, doi:10.4230/LIPIcs.SoCG.2016.51.
short: I. Mabillard, U. Wagner, in:, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik
GmbH, 2016, p. 51.1-51.12.
conference:
end_date: 2016-06-17
location: Medford, MA, USA
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2016-06-14
date_created: 2018-12-11T11:51:41Z
date_published: 2016-06-01T00:00:00Z
date_updated: 2021-01-12T06:50:17Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SoCG.2016.51
file:
- access_level: open_access
checksum: 92c0c3735fe908f8ded6e484005cb3b1
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:06Z
date_updated: 2020-07-14T12:44:47Z
file_id: '4791'
file_name: IST-2016-621-v1+1_LIPIcs-SoCG-2016-51.pdf
file_size: 622969
relation: main_file
file_date_updated: 2020-07-14T12:44:47Z
has_accepted_license: '1'
intvolume: ' 51'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 51.1 - 51.12
project:
- _id: 25FA3206-B435-11E9-9278-68D0E5697425
grant_number: PP00P2_138948
name: 'Embeddings in Higher Dimensions: Algorithms and Combinatorics'
publication_status: published
publisher: Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH
publist_id: '5830'
pubrep_id: '621'
quality_controlled: '1'
scopus_import: 1
status: public
title: Eliminating higher-multiplicity intersections, II. The deleted product criterion
in the r-metastable range
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: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 51
year: '2016'
...
---
_id: '1408'
abstract:
- lang: eng
text: 'The concept of well group in a special but important case captures homological
properties of the zero set of a continuous map (Formula presented.) on a compact
space K that are invariant with respect to perturbations of f. The perturbations
are arbitrary continuous maps within (Formula presented.) distance r from f for
a given (Formula presented.). The main drawback of the approach is that the computability
of well groups was shown only when (Formula presented.) or (Formula presented.).
Our contribution to the theory of well groups is twofold: on the one hand we improve
on the computability issue, but on the other hand we present a range of examples
where the well groups are incomplete invariants, that is, fail to capture certain
important robust properties of the zero set. For the first part, we identify a
computable subgroup of the well group that is obtained by cap product with the
pullback of the orientation of (Formula presented.) by f. In other words, well
groups can be algorithmically approximated from below. When f is smooth and (Formula
presented.), our approximation of the (Formula presented.)th well group is exact.
For the second part, we find examples of maps (Formula presented.) with all well
groups isomorphic but whose perturbations have different zero sets. We discuss
on a possible replacement of the well groups of vector valued maps by an invariant
of a better descriptive power and computability status.'
acknowledgement: 'Open access funding provided by Institute of Science and Technology
(IST Austria). '
article_processing_charge: Yes (via OA deal)
author:
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
citation:
ama: Franek P, Krcál M. On computability and triviality of well groups. Discrete
& Computational Geometry. 2016;56(1):126-164. doi:10.1007/s00454-016-9794-2
apa: Franek, P., & Krcál, M. (2016). On computability and triviality of well
groups. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/s00454-016-9794-2
chicago: Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well
Groups.” Discrete & Computational Geometry. Springer, 2016. https://doi.org/10.1007/s00454-016-9794-2.
ieee: P. Franek and M. Krcál, “On computability and triviality of well groups,”
Discrete & Computational Geometry, vol. 56, no. 1. Springer, pp. 126–164,
2016.
ista: Franek P, Krcál M. 2016. On computability and triviality of well groups. Discrete
& Computational Geometry. 56(1), 126–164.
mla: Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well Groups.”
Discrete & Computational Geometry, vol. 56, no. 1, Springer, 2016,
pp. 126–64, doi:10.1007/s00454-016-9794-2.
short: P. Franek, M. Krcál, Discrete & Computational Geometry 56 (2016) 126–164.
date_created: 2018-12-11T11:51:51Z
date_published: 2016-07-01T00:00:00Z
date_updated: 2023-02-23T10:02:11Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/s00454-016-9794-2
ec_funded: 1
file:
- access_level: open_access
checksum: e0da023abf6b72abd8c6a8c76740d53c
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:55Z
date_updated: 2020-07-14T12:44:53Z
file_id: '4846'
file_name: IST-2016-614-v1+1_s00454-016-9794-2.pdf
file_size: 905303
relation: main_file
file_date_updated: 2020-07-14T12:44:53Z
has_accepted_license: '1'
intvolume: ' 56'
issue: '1'
language:
- iso: eng
month: '07'
oa: 1
oa_version: Published Version
page: 126 - 164
project:
- _id: 25F8B9BC-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: M01980
name: Robust invariants of Nonlinear Systems
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
name: IST Austria Open Access Fund
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5799'
pubrep_id: '614'
quality_controlled: '1'
related_material:
record:
- id: '1510'
relation: earlier_version
status: public
scopus_import: 1
status: public
title: On computability and triviality of well groups
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: 56
year: '2016'
...
---
_id: '1411'
abstract:
- lang: eng
text: We consider two systems (α1, …, αm) and (β1, …,βn) of simple curves drawn
on a compact two-dimensional surface M with boundary. Each αi and each βj is either
an arc meeting the boundary of M at its two endpoints, or a closed curve. The
αi are pairwise disjoint except for possibly sharing endpoints, and similarly
for the βj. We want to “untangle” the βj from the ai by a self-homeomorphism of
M; more precisely, we seek a homeomorphism φ:M→M fixing the boundary of M pointwise
such that the total number of crossings of the ai with the φ(βj) is as small as
possible. This problem is motivated by an application in the algorithmic theory
of embeddings and 3-manifolds. We prove that if M is planar, i.e., a sphere with
h ≥ 0 boundary components (“holes”), then O(mn) crossings can be achieved (independently
of h), which is asymptotically tight, as an easy lower bound shows. In general,
for an arbitrary (orientable or nonorientable) surface M with h holes and of (orientable
or nonorientable) genus g ≥ 0, we obtain an O((m + n)4) upper bound, again independent
of h and g. The proofs rely, among other things, on a result concerning simultaneous
planar drawings of graphs by Erten and Kobourov.
acknowledgement: 'Supported by the ERC Adv anced Grant No. 267165. '
author:
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Eric
full_name: Sedgwick, Eric
last_name: Sedgwick
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Matoušek J, Sedgwick E, Tancer M, Wagner U. Untangling two systems of noncrossing
curves. Israel Journal of Mathematics. 2016;212(1):37-79. doi:10.1007/s11856-016-1294-9
apa: Matoušek, J., Sedgwick, E., Tancer, M., & Wagner, U. (2016). Untangling
two systems of noncrossing curves. Israel Journal of Mathematics. Springer.
https://doi.org/10.1007/s11856-016-1294-9
chicago: Matoušek, Jiří, Eric Sedgwick, Martin Tancer, and Uli Wagner. “Untangling
Two Systems of Noncrossing Curves.” Israel Journal of Mathematics. Springer,
2016. https://doi.org/10.1007/s11856-016-1294-9.
ieee: J. Matoušek, E. Sedgwick, M. Tancer, and U. Wagner, “Untangling two systems
of noncrossing curves,” Israel Journal of Mathematics, vol. 212, no. 1.
Springer, pp. 37–79, 2016.
ista: Matoušek J, Sedgwick E, Tancer M, Wagner U. 2016. Untangling two systems of
noncrossing curves. Israel Journal of Mathematics. 212(1), 37–79.
mla: Matoušek, Jiří, et al. “Untangling Two Systems of Noncrossing Curves.” Israel
Journal of Mathematics, vol. 212, no. 1, Springer, 2016, pp. 37–79, doi:10.1007/s11856-016-1294-9.
short: J. Matoušek, E. Sedgwick, M. Tancer, U. Wagner, Israel Journal of Mathematics
212 (2016) 37–79.
date_created: 2018-12-11T11:51:52Z
date_published: 2016-05-01T00:00:00Z
date_updated: 2023-02-23T10:34:31Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s11856-016-1294-9
intvolume: ' 212'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1302.6475
month: '05'
oa: 1
oa_version: Preprint
page: 37 - 79
project:
- _id: 25FA3206-B435-11E9-9278-68D0E5697425
grant_number: PP00P2_138948
name: 'Embeddings in Higher Dimensions: Algorithms and Combinatorics'
publication: Israel Journal of Mathematics
publication_status: published
publisher: Springer
publist_id: '5796'
quality_controlled: '1'
related_material:
record:
- id: '2244'
relation: earlier_version
status: public
scopus_import: 1
status: public
title: Untangling two systems of noncrossing curves
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 212
year: '2016'
...
---
_id: '1237'
abstract:
- lang: eng
text: 'Bitmap images of arbitrary dimension may be formally perceived as unions
of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology
and homology groups are well known topological invariants of such sets. Cohomological
operations, such as the cup product, provide higher-order algebraic topological
invariants, especially important for digital images of dimension higher than 3.
If such an operation is determined at the level of simplicial chains [see e.g.
González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively
computable. However, decomposing a cubical complex into a simplicial one deleteriously
affects the efficiency of such an approach. In order to avoid this overhead, a
direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015,
253-275] for the cup product in cohomology, and implemented in the ChainCon software
package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for
the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series,
1947, 290-320] directly at the level of cubical chains, and we prove the correctness
of this formula. An implementation of this formula is programmed in C++ within
the ChainCon software framework. We provide a few examples and discuss the effectiveness
of this approach. One specific application follows from the fact that Steenrod
squares yield tests for the topological extension problem: Can a given map A →
Sd to a sphere Sd be extended to a given super-complex X of A? In particular,
the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value
r > 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the
extension problem.'
acknowledgement: The research conducted by both authors has received funding from
the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme (FP7/2007-2013) under REA grant agreements no. 291734 (for M. K.) and
no. 622033 (for P. P.).
alternative_title:
- LNCS
author:
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Pawel
full_name: Pilarczyk, Pawel
id: 3768D56A-F248-11E8-B48F-1D18A9856A87
last_name: Pilarczyk
citation:
ama: 'Krcál M, Pilarczyk P. Computation of cubical Steenrod squares. In: Vol 9667.
Springer; 2016:140-151. doi:10.1007/978-3-319-39441-1_13'
apa: 'Krcál, M., & Pilarczyk, P. (2016). Computation of cubical Steenrod squares
(Vol. 9667, pp. 140–151). Presented at the CTIC: Computational Topology in Image
Context, Marseille, France: Springer. https://doi.org/10.1007/978-3-319-39441-1_13'
chicago: Krcál, Marek, and Pawel Pilarczyk. “Computation of Cubical Steenrod Squares,”
9667:140–51. Springer, 2016. https://doi.org/10.1007/978-3-319-39441-1_13.
ieee: 'M. Krcál and P. Pilarczyk, “Computation of cubical Steenrod squares,” presented
at the CTIC: Computational Topology in Image Context, Marseille, France, 2016,
vol. 9667, pp. 140–151.'
ista: 'Krcál M, Pilarczyk P. 2016. Computation of cubical Steenrod squares. CTIC:
Computational Topology in Image Context, LNCS, vol. 9667, 140–151.'
mla: Krcál, Marek, and Pawel Pilarczyk. Computation of Cubical Steenrod Squares.
Vol. 9667, Springer, 2016, pp. 140–51, doi:10.1007/978-3-319-39441-1_13.
short: M. Krcál, P. Pilarczyk, in:, Springer, 2016, pp. 140–151.
conference:
end_date: 2016-06-17
location: Marseille, France
name: 'CTIC: Computational Topology in Image Context'
start_date: 2016-06-15
date_created: 2018-12-11T11:50:52Z
date_published: 2016-06-02T00:00:00Z
date_updated: 2021-01-12T06:49:18Z
day: '02'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/978-3-319-39441-1_13
ec_funded: 1
intvolume: ' 9667'
language:
- iso: eng
month: '06'
oa_version: None
page: 140 - 151
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
- _id: 255F06BE-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '622033'
name: Persistent Homology - Images, Data and Maps
publication_status: published
publisher: Springer
publist_id: '6096'
quality_controlled: '1'
scopus_import: 1
status: public
title: Computation of cubical Steenrod squares
type: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 9667
year: '2016'
...
---
_id: '1282'
abstract:
- lang: eng
text: 'We consider higher-dimensional generalizations of the normalized Laplacian
and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam
model Xk(n, p) of random k-dimensional simplicial complexes on n vertices. We
show that for p = Ω(logn/n), the eigenvalues of each of the matrices are a.a.s.
concentrated around two values. The main tool, which goes back to the work of
Garland, are arguments that relate the eigenvalues of these matrices to those
of graphs that arise as links of (k - 2)-dimensional faces. Garland’s result concerns
the Laplacian; we develop an analogous result for the adjacency matrix. The same
arguments apply to other models of random complexes which allow for dependencies
between the choices of k-dimensional simplices. In the second part of the paper,
we apply this to the question of possible higher-dimensional analogues of the
discrete Cheeger inequality, which in the classical case of graphs relates the
eigenvalues of a graph and its edge expansion. It is very natural to ask whether
this generalizes to higher dimensions and, in particular, whether the eigenvalues
of the higher-dimensional Laplacian capture the notion of coboundary expansion—a
higher-dimensional generalization of edge expansion that arose in recent work
of Linial and Meshulam and of Gromov; this question was raised, for instance,
by Dotterrer and Kahle. We show that this most straightforward version of a higher-dimensional
discrete Cheeger inequality fails, in quite a strong way: For every k ≥ 2 and
n ∈ N, there is a k-dimensional complex Yn k on n vertices that has strong spectral
expansion properties (all nontrivial eigenvalues of the normalised k-dimensional
Laplacian lie in the interval [1−O(1/√1), 1+0(1/√1]) but whose coboundary expansion
is bounded from above by O(log n/n) and so tends to zero as n → ∞; moreover, Yn
k can be taken to have vanishing integer homology in dimension less than k.'
author:
- first_name: Anna
full_name: Gundert, Anna
last_name: Gundert
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Gundert A, Wagner U. On eigenvalues of random complexes. Israel Journal
of Mathematics. 2016;216(2):545-582. doi:10.1007/s11856-016-1419-1
apa: Gundert, A., & Wagner, U. (2016). On eigenvalues of random complexes. Israel
Journal of Mathematics. Springer. https://doi.org/10.1007/s11856-016-1419-1
chicago: Gundert, Anna, and Uli Wagner. “On Eigenvalues of Random Complexes.” Israel
Journal of Mathematics. Springer, 2016. https://doi.org/10.1007/s11856-016-1419-1.
ieee: A. Gundert and U. Wagner, “On eigenvalues of random complexes,” Israel
Journal of Mathematics, vol. 216, no. 2. Springer, pp. 545–582, 2016.
ista: Gundert A, Wagner U. 2016. On eigenvalues of random complexes. Israel Journal
of Mathematics. 216(2), 545–582.
mla: Gundert, Anna, and Uli Wagner. “On Eigenvalues of Random Complexes.” Israel
Journal of Mathematics, vol. 216, no. 2, Springer, 2016, pp. 545–82, doi:10.1007/s11856-016-1419-1.
short: A. Gundert, U. Wagner, Israel Journal of Mathematics 216 (2016) 545–582.
date_created: 2018-12-11T11:51:07Z
date_published: 2016-10-01T00:00:00Z
date_updated: 2021-01-12T06:49:36Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s11856-016-1419-1
intvolume: ' 216'
issue: '2'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1411.4906
month: '10'
oa: 1
oa_version: Preprint
page: 545 - 582
publication: Israel Journal of Mathematics
publication_status: published
publisher: Springer
publist_id: '6034'
quality_controlled: '1'
scopus_import: 1
status: public
title: On eigenvalues of random complexes
type: journal_article
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 216
year: '2016'
...
---
_id: '8183'
abstract:
- lang: eng
text: "We study conditions under which a finite simplicial complex $K$ can be mapped
to $\\mathbb R^d$ without higher-multiplicity intersections. An almost $r$-embedding
is a map $f: K\\to \\mathbb R^d$ such that the images of any $r$\r\npairwise disjoint
simplices of $K$ do not have a common point. We show that if $r$ is not a prime
power and $d\\geq 2r+1$, then there is a counterexample to the topological Tverberg
conjecture, i.e., there is an almost $r$-embedding of\r\nthe $(d+1)(r-1)$-simplex
in $\\mathbb R^d$. This improves on previous constructions of counterexamples
(for $d\\geq 3r$) based on a series of papers by M. \\\"Ozaydin, M. Gromov, P.
Blagojevi\\'c, F. Frick, G. Ziegler, and the second and fourth present authors.
The counterexamples are obtained by proving the following algebraic criterion
in codimension 2: If $r\\ge3$ and if $K$ is a finite $2(r-1)$-complex then there
exists an almost $r$-embedding $K\\to \\mathbb R^{2r}$ if and only if there exists
a general position PL map $f:K\\to \\mathbb R^{2r}$ such that the algebraic intersection
number of the $f$-images of any $r$ pairwise disjoint simplices of $K$ is zero.
This result can be restated in terms of cohomological obstructions or equivariant
maps, and extends an analogous codimension 3 criterion by the second and fourth
authors. As another application we classify ornaments $f:S^3 \\sqcup S^3\\sqcup
S^3\\to \\mathbb R^5$ up to ornament\r\nconcordance. It follows from work of M.
Freedman, V. Krushkal and P. Teichner that the analogous criterion for $r=2$ is
false. We prove a lemma on singular higher-dimensional Borromean rings, yielding
an elementary proof of the counterexample."
acknowledgement: We would like to thank A. Klyachko, V. Krushkal, S. Melikhov, M.
Tancer, P. Teichner and anonymous referees for helpful discussions.
article_number: '1511.03501'
article_processing_charge: No
arxiv: 1
author:
- first_name: Sergey
full_name: Avvakumov, Sergey
id: 3827DAC8-F248-11E8-B48F-1D18A9856A87
last_name: Avvakumov
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: A.
full_name: Skopenkov, A.
last_name: Skopenkov
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2. arXiv.
apa: Avvakumov, S., Mabillard, I., Skopenkov, A., & Wagner, U. (n.d.). Eliminating
higher-multiplicity intersections, III. Codimension 2. arXiv.
chicago: Avvakumov, Sergey, Isaac Mabillard, A. Skopenkov, and Uli Wagner. “Eliminating
Higher-Multiplicity Intersections, III. Codimension 2.” ArXiv, n.d.
ieee: S. Avvakumov, I. Mabillard, A. Skopenkov, and U. Wagner, “Eliminating higher-multiplicity
intersections, III. Codimension 2,” arXiv. .
ista: Avvakumov S, Mabillard I, Skopenkov A, Wagner U. Eliminating higher-multiplicity
intersections, III. Codimension 2. arXiv, 1511.03501.
mla: Avvakumov, Sergey, et al. “Eliminating Higher-Multiplicity Intersections, III.
Codimension 2.” ArXiv, 1511.03501.
short: S. Avvakumov, I. Mabillard, A. Skopenkov, U. Wagner, ArXiv (n.d.).
date_created: 2020-07-30T10:45:19Z
date_published: 2015-11-15T00:00:00Z
date_updated: 2023-09-07T13:12:17Z
day: '15'
department:
- _id: UlWa
external_id:
arxiv:
- '1511.03501'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/1511.03501
month: '11'
oa: 1
oa_version: Preprint
publication: arXiv
publication_status: submitted
related_material:
record:
- id: '9308'
relation: later_version
status: public
- id: '10220'
relation: later_version
status: public
- id: '8156'
relation: dissertation_contains
status: public
status: public
title: Eliminating higher-multiplicity intersections, III. Codimension 2
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2015'
...
---
_id: '1642'
abstract:
- lang: eng
text: The Hanani-Tutte theorem is a classical result proved for the first time in
the 1930s that characterizes planar graphs as graphs that admit a drawing in the
plane in which every pair of edges not sharing a vertex cross an even number of
times. We generalize this result to clustered graphs with two disjoint clusters,
and show that a straightforward extension to flat clustered graphs with three
or more disjoint clusters is not possible. For general clustered graphs we show
a variant of the Hanani-Tutte theorem in the case when each cluster induces a
connected subgraph. Di Battista and Frati proved that clustered planarity of embedded
clustered graphs whose every face is incident to at most five vertices can be
tested in polynomial time. We give a new and short proof of this result, using
the matroid intersection algorithm.
acknowledgement: e research leading to these results has received funding fromthe
People Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme(FP7/2007-2013) under REA grant agreement no [291734], and ESF Eurogiga
project GraDR as GAˇCRGIG/11/E023.
article_number: 'P4.24 '
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
- first_name: Igor
full_name: Malinovič, Igor
last_name: Malinovič
- first_name: Dömötör
full_name: Pálvölgyi, Dömötör
last_name: Pálvölgyi
citation:
ama: Fulek R, Kynčl J, Malinovič I, Pálvölgyi D. Clustered planarity testing revisited.
Electronic Journal of Combinatorics. 2015;22(4). doi:10.37236/5002
apa: Fulek, R., Kynčl, J., Malinovič, I., & Pálvölgyi, D. (2015). Clustered
planarity testing revisited. Electronic Journal of Combinatorics. Electronic
Journal of Combinatorics. https://doi.org/10.37236/5002
chicago: Fulek, Radoslav, Jan Kynčl, Igor Malinovič, and Dömötör Pálvölgyi. “Clustered
Planarity Testing Revisited.” Electronic Journal of Combinatorics. Electronic
Journal of Combinatorics, 2015. https://doi.org/10.37236/5002.
ieee: R. Fulek, J. Kynčl, I. Malinovič, and D. Pálvölgyi, “Clustered planarity testing
revisited,” Electronic Journal of Combinatorics, vol. 22, no. 4. Electronic
Journal of Combinatorics, 2015.
ista: Fulek R, Kynčl J, Malinovič I, Pálvölgyi D. 2015. Clustered planarity testing
revisited. Electronic Journal of Combinatorics. 22(4), P4.24.
mla: Fulek, Radoslav, et al. “Clustered Planarity Testing Revisited.” Electronic
Journal of Combinatorics, vol. 22, no. 4, P4.24, Electronic Journal of Combinatorics,
2015, doi:10.37236/5002.
short: R. Fulek, J. Kynčl, I. Malinovič, D. Pálvölgyi, Electronic Journal of Combinatorics
22 (2015).
date_created: 2018-12-11T11:53:12Z
date_published: 2015-11-13T00:00:00Z
date_updated: 2023-02-21T16:03:02Z
day: '13'
ddc:
- '514'
- '516'
department:
- _id: UlWa
doi: 10.37236/5002
ec_funded: 1
external_id:
arxiv:
- '1305.4519'
file:
- access_level: open_access
checksum: 40b5920b49ee736694f59f39588ee206
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:15:03Z
date_updated: 2020-07-14T12:45:08Z
file_id: '5120'
file_name: IST-2016-714-v1+1_5002-15499-3-PB.pdf
file_size: 443655
relation: main_file
file_date_updated: 2020-07-14T12:45:08Z
has_accepted_license: '1'
intvolume: ' 22'
issue: '4'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Published Version
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication: Electronic Journal of Combinatorics
publication_identifier:
eissn:
- 1077-8926
publication_status: published
publisher: Electronic Journal of Combinatorics
publist_id: '5511'
pubrep_id: '714'
quality_controlled: '1'
related_material:
record:
- id: '10793'
relation: earlier_version
status: public
scopus_import: '1'
status: public
title: Clustered planarity testing revisited
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 22
year: '2015'
...
---
_id: '1682'
abstract:
- lang: eng
text: 'We study the problem of robust satisfiability of systems of nonlinear equations,
namely, whether for a given continuous function f:K→ ℝn on a finite simplicial
complex K and α > 0, it holds that each function g: K → ℝn such that ||g -
f || ∞ < α, has a root in K. Via a reduction to the extension problem of maps
into a sphere, we particularly show that this problem is decidable in polynomial
time for every fixed n, assuming dimK ≤ 2n - 3. This is a substantial extension
of previous computational applications of topological degree and related concepts
in numerical and interval analysis. Via a reverse reduction, we prove that the
problem is undecidable when dim K > 2n - 2, where the threshold comes from
the stable range in homotopy theory. For the lucidity of our exposition, we focus
on the setting when f is simplexwise linear. Such functions can approximate general
continuous functions, and thus we get approximation schemes and undecidability
of the robust satisfiability in other possible settings.'
article_number: '26'
author:
- first_name: Peter
full_name: Franek, Peter
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
citation:
ama: Franek P, Krcál M. Robust satisfiability of systems of equations. Journal
of the ACM. 2015;62(4). doi:10.1145/2751524
apa: Franek, P., & Krcál, M. (2015). Robust satisfiability of systems of equations.
Journal of the ACM. ACM. https://doi.org/10.1145/2751524
chicago: Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.”
Journal of the ACM. ACM, 2015. https://doi.org/10.1145/2751524.
ieee: P. Franek and M. Krcál, “Robust satisfiability of systems of equations,” Journal
of the ACM, vol. 62, no. 4. ACM, 2015.
ista: Franek P, Krcál M. 2015. Robust satisfiability of systems of equations. Journal
of the ACM. 62(4), 26.
mla: Franek, Peter, and Marek Krcál. “Robust Satisfiability of Systems of Equations.”
Journal of the ACM, vol. 62, no. 4, 26, ACM, 2015, doi:10.1145/2751524.
short: P. Franek, M. Krcál, Journal of the ACM 62 (2015).
date_created: 2018-12-11T11:53:27Z
date_published: 2015-08-01T00:00:00Z
date_updated: 2021-01-12T06:52:30Z
day: '01'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1145/2751524
intvolume: ' 62'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1402.0858
month: '08'
oa: 1
oa_version: Preprint
publication: Journal of the ACM
publication_status: published
publisher: ACM
publist_id: '5466'
quality_controlled: '1'
scopus_import: 1
status: public
title: Robust satisfiability of systems of equations
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 62
year: '2015'
...
---
_id: '1685'
abstract:
- lang: eng
text: "Given a graph G cellularly embedded on a surface Σ of genus g, a cut graph
is a subgraph of G such that cutting Σ along G yields a topological disk. We provide
a fixed parameter tractable approximation scheme for the problem of computing
the shortest cut graph, that is, for any ε > 0, we show how to compute a (1 + ε)
approximation of the shortest cut graph in time f(ε, g)n3.\r\nOur techniques first
rely on the computation of a spanner for the problem using the technique of brick
decompositions, to reduce the problem to the case of bounded tree-width. Then,
to solve the bounded tree-width case, we introduce a variant of the surface-cut
decomposition of Rué, Sau and Thilikos, which may be of independent interest."
alternative_title:
- LNCS
author:
- first_name: Vincent
full_name: Cohen Addad, Vincent
last_name: Cohen Addad
- first_name: Arnaud N
full_name: De Mesmay, Arnaud N
id: 3DB2F25C-F248-11E8-B48F-1D18A9856A87
last_name: De Mesmay
citation:
ama: 'Cohen Addad V, de Mesmay AN. A fixed parameter tractable approximation scheme
for the optimal cut graph of a surface. In: Vol 9294. Springer; 2015:386-398.
doi:10.1007/978-3-662-48350-3_33'
apa: 'Cohen Addad, V., & de Mesmay, A. N. (2015). A fixed parameter tractable
approximation scheme for the optimal cut graph of a surface (Vol. 9294, pp. 386–398).
Presented at the ESA: European Symposium on Algorithms, Patras, Greece: Springer.
https://doi.org/10.1007/978-3-662-48350-3_33'
chicago: Cohen Addad, Vincent, and Arnaud N de Mesmay. “A Fixed Parameter Tractable
Approximation Scheme for the Optimal Cut Graph of a Surface,” 9294:386–98. Springer,
2015. https://doi.org/10.1007/978-3-662-48350-3_33.
ieee: 'V. Cohen Addad and A. N. de Mesmay, “A fixed parameter tractable approximation
scheme for the optimal cut graph of a surface,” presented at the ESA: European
Symposium on Algorithms, Patras, Greece, 2015, vol. 9294, pp. 386–398.'
ista: 'Cohen Addad V, de Mesmay AN. 2015. A fixed parameter tractable approximation
scheme for the optimal cut graph of a surface. ESA: European Symposium on Algorithms,
LNCS, vol. 9294, 386–398.'
mla: Cohen Addad, Vincent, and Arnaud N. de Mesmay. A Fixed Parameter Tractable
Approximation Scheme for the Optimal Cut Graph of a Surface. Vol. 9294, Springer,
2015, pp. 386–98, doi:10.1007/978-3-662-48350-3_33.
short: V. Cohen Addad, A.N. de Mesmay, in:, Springer, 2015, pp. 386–398.
conference:
end_date: 2015-09-16
location: Patras, Greece
name: 'ESA: European Symposium on Algorithms'
start_date: 2015-09-14
date_created: 2018-12-11T11:53:27Z
date_published: 2015-09-01T00:00:00Z
date_updated: 2021-01-12T06:52:31Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/978-3-662-48350-3_33
ec_funded: 1
intvolume: ' 9294'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1507.01688
month: '09'
oa: 1
oa_version: Preprint
page: 386 - 398
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication_status: published
publisher: Springer
publist_id: '5462'
quality_controlled: '1'
scopus_import: 1
status: public
title: A fixed parameter tractable approximation scheme for the optimal cut graph
of a surface
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 9294
year: '2015'
...
---
_id: '1688'
abstract:
- lang: eng
text: 'We estimate the selection constant in the following geometric selection theorem
by Pach: For every positive integer d, there is a constant (Formula presented.)
such that whenever (Formula presented.) are n-element subsets of (Formula presented.),
we can find a point (Formula presented.) and subsets (Formula presented.) for
every i∈[d+1], each of size at least cdn, such that p belongs to all rainbowd-simplices
determined by (Formula presented.) simplices with one vertex in each Yi. We show
a super-exponentially decreasing upper bound (Formula presented.). The ideas used
in the proof of the upper bound also help us to prove Pach’s theorem with (Formula
presented.), which is a lower bound doubly exponentially decreasing in d (up to
some polynomial in the exponent). For comparison, Pach’s original approach yields
a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and
Suk recently obtained a hypergraph density result implying a proof of Pach’s theorem
with (Formula presented.). In our construction for the upper bound, we use the
fact that the minimum solid angle of every d-simplex is super-exponentially small.
This fact was previously unknown and might be of independent interest. For the
lower bound, we improve the ‘separation’ part of the argument by showing that
in one of the key steps only d+1 separations are necessary, compared to 2d separations
in the original proof. We also provide a measure version of Pach’s theorem.'
acknowledgement: R. K. was supported by the Russian Foundation for Basic Research
Grant 15-31-20403 (mol_a_ved) and grant 15-01-99563. J. K., Z. P., and M. T. were
partially supported by ERC Advanced Research Grant No. 267165 (DISCONV) and by the
project CE-ITI (GAČR P202/12/G061) of the Czech Science Foundation. J. K. was also
partially supported by Swiss National Science Foundation Grants 200021-137574 and
200020-14453. P. P., Z. P., and M. T. were partially supported by the Charles University
Grant GAUK 421511. P. P. was also partially supported by the Charles University
Grant SVV-2014-260107. Z. P. was also partially supported by the Charles University
Grant SVV-2014-260103.
author:
- first_name: Roman
full_name: Karasev, Roman
last_name: Karasev
- first_name: Jan
full_name: Kynčl, Jan
last_name: Kynčl
- first_name: Pavel
full_name: Paták, Pavel
last_name: Paták
- first_name: Zuzana
full_name: Patakova, Zuzana
last_name: Patakova
orcid: 0000-0002-3975-1683
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
citation:
ama: Karasev R, Kynčl J, Paták P, Patakova Z, Tancer M. Bounds for Pach’s selection
theorem and for the minimum solid angle in a simplex. Discrete & Computational
Geometry. 2015;54(3):610-636. doi:10.1007/s00454-015-9720-z
apa: Karasev, R., Kynčl, J., Paták, P., Patakova, Z., & Tancer, M. (2015). Bounds
for Pach’s selection theorem and for the minimum solid angle in a simplex. Discrete
& Computational Geometry. Springer. https://doi.org/10.1007/s00454-015-9720-z
chicago: Karasev, Roman, Jan Kynčl, Pavel Paták, Zuzana Patakova, and Martin Tancer.
“Bounds for Pach’s Selection Theorem and for the Minimum Solid Angle in a Simplex.”
Discrete & Computational Geometry. Springer, 2015. https://doi.org/10.1007/s00454-015-9720-z.
ieee: R. Karasev, J. Kynčl, P. Paták, Z. Patakova, and M. Tancer, “Bounds for Pach’s
selection theorem and for the minimum solid angle in a simplex,” Discrete &
Computational Geometry, vol. 54, no. 3. Springer, pp. 610–636, 2015.
ista: Karasev R, Kynčl J, Paták P, Patakova Z, Tancer M. 2015. Bounds for Pach’s
selection theorem and for the minimum solid angle in a simplex. Discrete &
Computational Geometry. 54(3), 610–636.
mla: Karasev, Roman, et al. “Bounds for Pach’s Selection Theorem and for the Minimum
Solid Angle in a Simplex.” Discrete & Computational Geometry, vol.
54, no. 3, Springer, 2015, pp. 610–36, doi:10.1007/s00454-015-9720-z.
short: R. Karasev, J. Kynčl, P. Paták, Z. Patakova, M. Tancer, Discrete & Computational
Geometry 54 (2015) 610–636.
date_created: 2018-12-11T11:53:28Z
date_published: 2015-10-01T00:00:00Z
date_updated: 2021-01-12T06:52:32Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-015-9720-z
intvolume: ' 54'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1403.8147
month: '10'
oa: 1
oa_version: Preprint
page: 610 - 636
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5457'
quality_controlled: '1'
scopus_import: 1
status: public
title: Bounds for Pach's selection theorem and for the minimum solid angle in a simplex
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 54
year: '2015'
...
---
_id: '1730'
abstract:
- lang: eng
text: How much cutting is needed to simplify the topology of a surface? We provide
bounds for several instances of this question, for the minimum length of topologically
non-trivial closed curves, pants decompositions, and cut graphs with a given combinatorial
map in triangulated combinatorial surfaces (or their dual cross-metric counterpart).
Our work builds upon Riemannian systolic inequalities, which bound the minimum
length of non-trivial closed curves in terms of the genus and the area of the
surface. We first describe a systematic way to translate Riemannian systolic inequalities
to a discrete setting, and vice-versa. This implies a conjecture by Przytycka
and Przytycki (Graph structure theory. Contemporary Mathematics, vol. 147, 1993),
a number of new systolic inequalities in the discrete setting, and the fact that
a theorem of Hutchinson on the edge-width of triangulated surfaces and Gromov’s
systolic inequality for surfaces are essentially equivalent. We also discuss how
these proofs generalize to higher dimensions. Then we focus on topological decompositions
of surfaces. Relying on ideas of Buser, we prove the existence of pants decompositions
of length O(g^(3/2)n^(1/2)) for any triangulated combinatorial surface of genus
g with n triangles, and describe an O(gn)-time algorithm to compute such a decomposition.
Finally, we consider the problem of embedding a cut graph (or more generally a
cellular graph) with a given combinatorial map on a given surface. Using random
triangulations, we prove (essentially) that, for any choice of a combinatorial
map, there are some surfaces on which any cellular embedding with that combinatorial
map has length superlinear in the number of triangles of the triangulated combinatorial
surface. There is also a similar result for graphs embedded on polyhedral triangulations.
author:
- first_name: Éric
full_name: Colin De Verdière, Éric
last_name: Colin De Verdière
- first_name: Alfredo
full_name: Hubard, Alfredo
last_name: Hubard
- first_name: Arnaud N
full_name: De Mesmay, Arnaud N
id: 3DB2F25C-F248-11E8-B48F-1D18A9856A87
last_name: De Mesmay
citation:
ama: Colin De Verdière É, Hubard A, de Mesmay AN. Discrete systolic inequalities
and decompositions of triangulated surfaces. Discrete & Computational Geometry.
2015;53(3):587-620. doi:10.1007/s00454-015-9679-9
apa: Colin De Verdière, É., Hubard, A., & de Mesmay, A. N. (2015). Discrete
systolic inequalities and decompositions of triangulated surfaces. Discrete
& Computational Geometry. Springer. https://doi.org/10.1007/s00454-015-9679-9
chicago: Colin De Verdière, Éric, Alfredo Hubard, and Arnaud N de Mesmay. “Discrete
Systolic Inequalities and Decompositions of Triangulated Surfaces.” Discrete
& Computational Geometry. Springer, 2015. https://doi.org/10.1007/s00454-015-9679-9.
ieee: É. Colin De Verdière, A. Hubard, and A. N. de Mesmay, “Discrete systolic inequalities
and decompositions of triangulated surfaces,” Discrete & Computational
Geometry, vol. 53, no. 3. Springer, pp. 587–620, 2015.
ista: Colin De Verdière É, Hubard A, de Mesmay AN. 2015. Discrete systolic inequalities
and decompositions of triangulated surfaces. Discrete & Computational Geometry.
53(3), 587–620.
mla: Colin De Verdière, Éric, et al. “Discrete Systolic Inequalities and Decompositions
of Triangulated Surfaces.” Discrete & Computational Geometry, vol.
53, no. 3, Springer, 2015, pp. 587–620, doi:10.1007/s00454-015-9679-9.
short: É. Colin De Verdière, A. Hubard, A.N. de Mesmay, Discrete & Computational
Geometry 53 (2015) 587–620.
date_created: 2018-12-11T11:53:42Z
date_published: 2015-04-02T00:00:00Z
date_updated: 2021-01-12T06:52:49Z
day: '02'
department:
- _id: UlWa
doi: 10.1007/s00454-015-9679-9
intvolume: ' 53'
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1408.4036
month: '04'
oa: 1
oa_version: Preprint
page: 587 - 620
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5397'
quality_controlled: '1'
scopus_import: 1
status: public
title: Discrete systolic inequalities and decompositions of triangulated surfaces
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 53
year: '2015'
...
---
_id: '1510'
abstract:
- lang: eng
text: 'The concept of well group in a special but important case captures homological
properties of the zero set of a continuous map f from K to R^n on a compact space
K that are invariant with respect to perturbations of f. The perturbations are
arbitrary continuous maps within L_infty distance r from f for a given r >
0. The main drawback of the approach is that the computability of well groups
was shown only when dim K = n or n = 1. Our contribution to the theory of well
groups is twofold: on the one hand we improve on the computability issue, but
on the other hand we present a range of examples where the well groups are incomplete
invariants, that is, fail to capture certain important robust properties of the
zero set. For the first part, we identify a computable subgroup of the well group
that is obtained by cap product with the pullback of the orientation of R^n by
f. In other words, well groups can be algorithmically approximated from below.
When f is smooth and dim K < 2n-2, our approximation of the (dim K-n)th well
group is exact. For the second part, we find examples of maps f, f'' from K to
R^n with all well groups isomorphic but whose perturbations have different zero
sets. We discuss on a possible replacement of the well groups of vector valued
maps by an invariant of a better descriptive power and computability status. '
alternative_title:
- LIPIcs
author:
- first_name: Peter
full_name: Franek, Peter
id: 473294AE-F248-11E8-B48F-1D18A9856A87
last_name: Franek
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
citation:
ama: 'Franek P, Krcál M. On computability and triviality of well groups. In: Vol
34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:842-856. doi:10.4230/LIPIcs.SOCG.2015.842'
apa: 'Franek, P., & Krcál, M. (2015). On computability and triviality of well
groups (Vol. 34, pp. 842–856). Presented at the SoCG: Symposium on Computational
Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik.
https://doi.org/10.4230/LIPIcs.SOCG.2015.842'
chicago: Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well
Groups,” 34:842–56. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. https://doi.org/10.4230/LIPIcs.SOCG.2015.842.
ieee: 'P. Franek and M. Krcál, “On computability and triviality of well groups,”
presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands,
2015, vol. 34, pp. 842–856.'
ista: 'Franek P, Krcál M. 2015. On computability and triviality of well groups.
SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 842–856.'
mla: Franek, Peter, and Marek Krcál. On Computability and Triviality of Well
Groups. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015,
pp. 842–56, doi:10.4230/LIPIcs.SOCG.2015.842.
short: P. Franek, M. Krcál, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2015, pp. 842–856.
conference:
end_date: 2015-06-25
location: Eindhoven, Netherlands
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2015-06-22
date_created: 2018-12-11T11:52:26Z
date_published: 2015-06-11T00:00:00Z
date_updated: 2023-02-21T17:02:57Z
day: '11'
ddc:
- '510'
department:
- _id: UlWa
- _id: HeEd
doi: 10.4230/LIPIcs.SOCG.2015.842
ec_funded: 1
file:
- access_level: open_access
checksum: 49eb5021caafaabe5356c65b9c5f8c9c
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:13:19Z
date_updated: 2020-07-14T12:44:59Z
file_id: '5001'
file_name: IST-2016-503-v1+1_32.pdf
file_size: 623563
relation: main_file
file_date_updated: 2020-07-14T12:44:59Z
has_accepted_license: '1'
intvolume: ' 34'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 842 - 856
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '5667'
pubrep_id: '503'
quality_controlled: '1'
related_material:
record:
- id: '1408'
relation: later_version
status: public
scopus_import: 1
status: public
title: On computability and triviality of well groups
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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2015'
...
---
_id: '1511'
abstract:
- lang: eng
text: 'The fact that the complete graph K_5 does not embed in the plane has been
generalized in two independent directions. On the one hand, the solution of the
classical Heawood problem for graphs on surfaces established that the complete
graph K_n embeds in a closed surface M if and only if (n-3)(n-4) is at most 6b_1(M),
where b_1(M) is the first Z_2-Betti number of M. On the other hand, Van Kampen
and Flores proved that the k-skeleton of the n-dimensional simplex (the higher-dimensional
analogue of K_{n+1}) embeds in R^{2k} if and only if n is less or equal to 2k+2.
Two decades ago, Kuhnel conjectured that the k-skeleton of the n-simplex embeds
in a compact, (k-1)-connected 2k-manifold with kth Z_2-Betti number b_k only if
the following generalized Heawood inequality holds: binom{n-k-1}{k+1} is at most
binom{2k+1}{k+1} b_k. This is a common generalization of the case of graphs on
surfaces as well as the Van Kampen--Flores theorem. In the spirit of Kuhnel''s
conjecture, we prove that if the k-skeleton of the n-simplex embeds in a 2k-manifold
with kth Z_2-Betti number b_k, then n is at most 2b_k binom{2k+2}{k} + 2k + 5.
This bound is weaker than the generalized Heawood inequality, but does not require
the assumption that M is (k-1)-connected. Our proof uses a result of Volovikov
about maps that satisfy a certain homological triviality condition.'
acknowledgement: "The work by Z. P. was partially supported by the Charles University
Grant SVV-2014-260103. The\r\nwork by Z. P. and M. T. was partially supported by
the project CE-ITI (GACR P202/12/G061) of\r\nthe Czech Science Foundation and by
the ERC Advanced Grant No. 267165. Part of the research\r\nwork of M. T. was conducted
at IST Austria, supported by an IST Fellowship. The work by U.W.\r\nwas partially
supported by the Swiss National Science Foundation (grants SNSF-200020-138230 and\r\nSNSF-PP00P2-138948)."
alternative_title:
- LIPIcs
author:
- first_name: Xavier
full_name: Goaoc, Xavier
last_name: Goaoc
- first_name: Isaac
full_name: Mabillard, Isaac
id: 32BF9DAA-F248-11E8-B48F-1D18A9856A87
last_name: Mabillard
- first_name: Pavel
full_name: Paták, Pavel
last_name: Paták
- 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
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. On generalized
Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability
result. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:476-490.
doi:10.4230/LIPIcs.SOCG.2015.476'
apa: 'Goaoc, X., Mabillard, I., Paták, P., Patakova, Z., Tancer, M., & Wagner,
U. (2015). On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type
nonembeddability result (Vol. 34, pp. 476–490). Presented at the SoCG: Symposium
on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum
für Informatik. https://doi.org/10.4230/LIPIcs.SOCG.2015.476'
chicago: 'Goaoc, Xavier, Isaac Mabillard, Pavel Paták, Zuzana Patakova, Martin Tancer,
and Uli Wagner. “On Generalized Heawood Inequalities for Manifolds: A Van Kampen–Flores-Type
Nonembeddability Result,” 34:476–90. Schloss Dagstuhl - Leibniz-Zentrum für Informatik,
2015. https://doi.org/10.4230/LIPIcs.SOCG.2015.476.'
ieee: 'X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, and U. Wagner,
“On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability
result,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven,
Netherlands, 2015, vol. 34, pp. 476–490.'
ista: 'Goaoc X, Mabillard I, Paták P, Patakova Z, Tancer M, Wagner U. 2015. On generalized
Heawood inequalities for manifolds: A Van Kampen–Flores-type nonembeddability
result. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 476–490.'
mla: 'Goaoc, Xavier, et al. On Generalized Heawood Inequalities for Manifolds:
A Van Kampen–Flores-Type Nonembeddability Result. Vol. 34, Schloss Dagstuhl
- Leibniz-Zentrum für Informatik, 2015, pp. 476–90, doi:10.4230/LIPIcs.SOCG.2015.476.'
short: X. Goaoc, I. Mabillard, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:,
Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 476–490.
conference:
end_date: 2015-06-25
location: Eindhoven, Netherlands
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2015-06-22
date_created: 2018-12-11T11:52:27Z
date_published: 2015-06-11T00:00:00Z
date_updated: 2023-02-23T12:38:00Z
day: '11'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SOCG.2015.476
ec_funded: 1
file:
- access_level: open_access
checksum: 0945811875351796324189312ca29e9e
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:11:18Z
date_updated: 2020-07-14T12:44:59Z
file_id: '4871'
file_name: IST-2016-502-v1+1_42.pdf
file_size: 636735
relation: main_file
file_date_updated: 2020-07-14T12:44:59Z
has_accepted_license: '1'
language:
- iso: eng
month: '06'
oa: 1
oa_version: Published Version
page: 476 - 490
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '5666'
pubrep_id: '502'
quality_controlled: '1'
related_material:
record:
- id: '610'
relation: later_version
status: public
scopus_import: 1
status: public
title: 'On generalized Heawood inequalities for manifolds: A Van Kampen–Flores-type
nonembeddability result'
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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: '34 '
year: '2015'
...
---
_id: '1512'
abstract:
- lang: eng
text: 'We show that very weak topological assumptions are enough to ensure the existence
of a Helly-type theorem. More precisely, we show that for any non-negative integers
b and d there exists an integer h(b,d) such that the following holds. If F is
a finite family of subsets of R^d such that the ith reduced Betti number (with
Z_2 coefficients in singular homology) of the intersection of any proper subfamily
G of F is at most b for every non-negative integer i less or equal to (d-1)/2,
then F has Helly number at most h(b,d). These topological conditions are sharp:
not controlling any of these first Betti numbers allow for families with unbounded
Helly number. Our proofs combine homological non-embeddability results with a
Ramsey-based approach to build, given an arbitrary simplicial complex K, some
well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent
interest.'
acknowledgement: "PP, ZP and MT were partially supported by the Charles University
Grant GAUK 421511. ZP was\r\npartially supported by the Charles University Grant
SVV-2014-260103. ZP and MT were partially\r\nsupported by the ERC Advanced Grant
No. 267165 and by the project CE-ITI (GACR P202/12/G061)\r\nof the Czech Science
Foundation. UW was partially supported by the Swiss National Science Foundation\r\n(grants
SNSF-200020-138230 and SNSF-PP00P2-138948). Part of this work was done when XG was
affiliated with INRIA Nancy Grand-Est and when MT was affiliated with Institutionen
för matematik, Kungliga Tekniska Högskolan, then IST Austria."
alternative_title:
- LIPIcs
article_processing_charge: No
author:
- first_name: Xavier
full_name: Goaoc, Xavier
last_name: Goaoc
- first_name: Pavel
full_name: Paták, Pavel
last_name: Paták
- first_name: Zuzana
full_name: Patakova, Zuzana
last_name: Patakova
orcid: 0000-0002-3975-1683
- first_name: Martin
full_name: Tancer, Martin
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. Bounding Helly numbers via
Betti numbers. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik;
2015:507-521. doi:10.4230/LIPIcs.SOCG.2015.507'
apa: 'Goaoc, X., Paták, P., Patakova, Z., Tancer, M., & Wagner, U. (2015). Bounding
Helly numbers via Betti numbers (Vol. 34, pp. 507–521). Presented at the SoCG:
Symposium on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl
- Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.SOCG.2015.507'
chicago: Goaoc, Xavier, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner.
“Bounding Helly Numbers via Betti Numbers,” 34:507–21. Schloss Dagstuhl - Leibniz-Zentrum
für Informatik, 2015. https://doi.org/10.4230/LIPIcs.SOCG.2015.507.
ieee: 'X. Goaoc, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “Bounding Helly
numbers via Betti numbers,” presented at the SoCG: Symposium on Computational
Geometry, Eindhoven, Netherlands, 2015, vol. 34, pp. 507–521.'
ista: 'Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. 2015. Bounding Helly numbers
via Betti numbers. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34,
507–521.'
mla: Goaoc, Xavier, et al. Bounding Helly Numbers via Betti Numbers. Vol.
34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 507–21, doi:10.4230/LIPIcs.SOCG.2015.507.
short: X. Goaoc, P. Paták, Z. Patakova, M. Tancer, U. Wagner, in:, Schloss Dagstuhl
- Leibniz-Zentrum für Informatik, 2015, pp. 507–521.
conference:
end_date: 2015-06-25
location: Eindhoven, Netherlands
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2015-06-22
date_created: 2018-12-11T11:52:27Z
date_published: 2015-01-01T00:00:00Z
date_updated: 2024-02-28T12:59:37Z
day: '01'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.4230/LIPIcs.SOCG.2015.507
file:
- access_level: open_access
checksum: e6881df44d87fe0c2529c9f7b2724614
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:10:09Z
date_updated: 2020-07-14T12:45:00Z
file_id: '4794'
file_name: IST-2016-501-v1+1_46.pdf
file_size: 633712
relation: main_file
file_date_updated: 2020-07-14T12:45:00Z
has_accepted_license: '1'
intvolume: ' 34'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Submitted Version
page: 507 - 521
publication_status: published
publisher: Schloss Dagstuhl - Leibniz-Zentrum für Informatik
publist_id: '5665'
pubrep_id: '501'
quality_controlled: '1'
related_material:
record:
- id: '424'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Bounding Helly numbers via Betti numbers
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: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 34
year: '2015'
...
---
_id: '1595'
abstract:
- lang: eng
text: 'A drawing of a graph G is radial if the vertices of G are placed on concentric
circles C1, . . . , Ck with common center c, and edges are drawn radially: every
edge intersects every circle centered at c at most once. G is radial planar if
it has a radial embedding, that is, a crossing- free radial drawing. If the vertices
of G are ordered or partitioned into ordered levels (as they are for leveled graphs),
we require that the assignment of vertices to circles corresponds to the given
ordering or leveling. We show that a graph G is radial planar if G has a radial
drawing in which every two edges cross an even number of times; the radial embedding
has the same leveling as the radial drawing. In other words, we establish the
weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes
a result by Pach and Tóth.'
acknowledgement: The research leading to these results has received funding from the
People Programme (Marie Curie Actions) of the European Union’s Seventh Framework
Programme (FP7/2007-2013) under REA grant agreement no [291734].
alternative_title:
- LNCS
author:
- first_name: Radoslav
full_name: Fulek, Radoslav
id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87
last_name: Fulek
orcid: 0000-0001-8485-1774
- first_name: Michael
full_name: Pelsmajer, Michael
last_name: Pelsmajer
- first_name: Marcus
full_name: Schaefer, Marcus
last_name: Schaefer
citation:
ama: 'Fulek R, Pelsmajer M, Schaefer M. Hanani-Tutte for radial planarity. In: Vol
9411. Springer; 2015:99-110. doi:10.1007/978-3-319-27261-0_9'
apa: 'Fulek, R., Pelsmajer, M., & Schaefer, M. (2015). Hanani-Tutte for radial
planarity (Vol. 9411, pp. 99–110). Presented at the GD: Graph Drawing and Network
Visualization, Los Angeles, CA, USA: Springer. https://doi.org/10.1007/978-3-319-27261-0_9'
chicago: Fulek, Radoslav, Michael Pelsmajer, and Marcus Schaefer. “Hanani-Tutte
for Radial Planarity,” 9411:99–110. Springer, 2015. https://doi.org/10.1007/978-3-319-27261-0_9.
ieee: 'R. Fulek, M. Pelsmajer, and M. Schaefer, “Hanani-Tutte for radial planarity,”
presented at the GD: Graph Drawing and Network Visualization, Los Angeles, CA,
USA, 2015, vol. 9411, pp. 99–110.'
ista: 'Fulek R, Pelsmajer M, Schaefer M. 2015. Hanani-Tutte for radial planarity.
GD: Graph Drawing and Network Visualization, LNCS, vol. 9411, 99–110.'
mla: Fulek, Radoslav, et al. Hanani-Tutte for Radial Planarity. Vol. 9411,
Springer, 2015, pp. 99–110, doi:10.1007/978-3-319-27261-0_9.
short: R. Fulek, M. Pelsmajer, M. Schaefer, in:, Springer, 2015, pp. 99–110.
conference:
end_date: 2015-09-26
location: Los Angeles, CA, USA
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2015-09-24
date_created: 2018-12-11T11:52:55Z
date_published: 2015-11-27T00:00:00Z
date_updated: 2023-02-21T16:23:36Z
day: '27'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/978-3-319-27261-0_9
ec_funded: 1
file:
- access_level: open_access
checksum: 685f91bd077a951ba067d42cce75409e
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:08:36Z
date_updated: 2020-07-14T12:45:03Z
file_id: '4697'
file_name: IST-2016-594-v1+1_HTCylinder_GD_Revision.pdf
file_size: 330135
relation: main_file
file_date_updated: 2020-07-14T12:45:03Z
has_accepted_license: '1'
intvolume: ' 9411'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Submitted Version
page: 99 - 110
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication_status: published
publisher: Springer
publist_id: '5576'
pubrep_id: '594'
quality_controlled: '1'
related_material:
record:
- id: '1113'
relation: later_version
status: public
- id: '1164'
relation: later_version
status: public
scopus_import: 1
status: public
title: Hanani-Tutte for radial planarity
type: conference
user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87
volume: 9411
year: '2015'
...
---
_id: '1596'
abstract:
- lang: eng
text: Let C={C1,...,Cn} denote a collection of translates of a regular convex k-gon
in the plane with the stacking order. The collection C forms a visibility clique
if for everyi < j the intersection Ci and (Ci ∩ Cj)\⋃i<l<jCl =∅.elements
that are stacked between them, i.e., We show that if C forms a visibility clique
its size is bounded from above by O(k4) thereby improving the upper bound of 22k
from the aforementioned paper. We also obtain an upper bound of 22(k/2)+2 on the
size of a visibility clique for homothetes of a convex (not necessarily regular)
k-gon.
alternative_title:
- LNCS
article_processing_charge: No
author:
- first_name: Radoslav
full_name: Fulek, Radoslav
id: 39F3FFE4-F248-11E8-B48F-1D18A9856A87
last_name: Fulek
orcid: 0000-0001-8485-1774
- first_name: Radoš
full_name: Radoičić, Radoš
last_name: Radoičić
citation:
ama: 'Fulek R, Radoičić R. Vertical visibility among parallel polygons in three
dimensions. In: Graph Drawing and Network Visualization. Vol 9411. Springer
Nature; 2015:373-379. doi:10.1007/978-3-319-27261-0_31'
apa: 'Fulek, R., & Radoičić, R. (2015). Vertical visibility among parallel polygons
in three dimensions. In Graph Drawing and Network Visualization (Vol. 9411,
pp. 373–379). Los Angeles, CA, United States: Springer Nature. https://doi.org/10.1007/978-3-319-27261-0_31'
chicago: Fulek, Radoslav, and Radoš Radoičić. “Vertical Visibility among Parallel
Polygons in Three Dimensions.” In Graph Drawing and Network Visualization,
9411:373–79. Springer Nature, 2015. https://doi.org/10.1007/978-3-319-27261-0_31.
ieee: R. Fulek and R. Radoičić, “Vertical visibility among parallel polygons in
three dimensions,” in Graph Drawing and Network Visualization, vol. 9411,
Springer Nature, 2015, pp. 373–379.
ista: 'Fulek R, Radoičić R. 2015.Vertical visibility among parallel polygons in
three dimensions. In: Graph Drawing and Network Visualization. LNCS, vol. 9411,
373–379.'
mla: Fulek, Radoslav, and Radoš Radoičić. “Vertical Visibility among Parallel Polygons
in Three Dimensions.” Graph Drawing and Network Visualization, vol. 9411,
Springer Nature, 2015, pp. 373–79, doi:10.1007/978-3-319-27261-0_31.
short: R. Fulek, R. Radoičić, in:, Graph Drawing and Network Visualization, Springer
Nature, 2015, pp. 373–379.
conference:
end_date: 2015-09-26
location: Los Angeles, CA, United States
name: 'GD: Graph Drawing and Network Visualization'
start_date: 2015-09-24
date_created: 2018-12-11T11:52:56Z
date_published: 2015-11-27T00:00:00Z
date_updated: 2022-01-28T09:20:50Z
day: '27'
ddc:
- '510'
department:
- _id: UlWa
doi: 10.1007/978-3-319-27261-0_31
ec_funded: 1
file:
- access_level: open_access
checksum: eec04f86c5921d04f025d5791db9b965
content_type: application/pdf
creator: system
date_created: 2018-12-12T10:17:06Z
date_updated: 2020-07-14T12:45:04Z
file_id: '5258'
file_name: IST-2016-595-v1+1_VerticalVisibilityGDRevision.pdf
file_size: 312992
relation: main_file
file_date_updated: 2020-07-14T12:45:04Z
has_accepted_license: '1'
intvolume: ' 9411'
language:
- iso: eng
month: '11'
oa: 1
oa_version: Submitted Version
page: 373 - 379
project:
- _id: 25681D80-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '291734'
name: International IST Postdoc Fellowship Programme
publication: Graph Drawing and Network Visualization
publication_identifier:
isbn:
- 978-3-319-27260-3
publication_status: published
publisher: Springer Nature
publist_id: '5575'
pubrep_id: '595'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Vertical visibility among parallel polygons in three dimensions
type: book_chapter
user_id: 8b945eb4-e2f2-11eb-945a-df72226e66a9
volume: 9411
year: '2015'
...
---
_id: '10793'
abstract:
- lang: eng
text: "The Hanani–Tutte theorem is a classical result proved for the first time
in the 1930s that characterizes planar graphs as graphs that admit a drawing in
the plane in which every pair of edges not sharing a vertex cross an even number
of times. We generalize this classical result to clustered graphs with two disjoint
clusters, and show that a straightforward extension of our result to flat clustered
graphs with three or more disjoint clusters is not possible.\r\n\r\nWe also give
a new and short proof for a related result by Di Battista and Frati based on the
matroid intersection algorithm."
alternative_title:
- LNCS
article_processing_charge: No
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
- first_name: Igor
full_name: Malinović, Igor
last_name: Malinović
- first_name: Dömötör
full_name: Pálvölgyi, Dömötör
last_name: Pálvölgyi
citation:
ama: 'Fulek R, Kynčl J, Malinović I, Pálvölgyi D. Clustered planarity testing revisited.
In: International Symposium on Graph Drawing. Vol 8871. Cham: Springer
Nature; 2014:428-436. doi:10.1007/978-3-662-45803-7_36'
apa: 'Fulek, R., Kynčl, J., Malinović, I., & Pálvölgyi, D. (2014). Clustered
planarity testing revisited. In International Symposium on Graph Drawing
(Vol. 8871, pp. 428–436). Cham: Springer Nature. https://doi.org/10.1007/978-3-662-45803-7_36'
chicago: 'Fulek, Radoslav, Jan Kynčl, Igor Malinović, and Dömötör Pálvölgyi. “Clustered
Planarity Testing Revisited.” In International Symposium on Graph Drawing,
8871:428–36. Cham: Springer Nature, 2014. https://doi.org/10.1007/978-3-662-45803-7_36.'
ieee: R. Fulek, J. Kynčl, I. Malinović, and D. Pálvölgyi, “Clustered planarity testing
revisited,” in International Symposium on Graph Drawing, 2014, vol. 8871,
pp. 428–436.
ista: Fulek R, Kynčl J, Malinović I, Pálvölgyi D. 2014. Clustered planarity testing
revisited. International Symposium on Graph Drawing. , LNCS, vol. 8871, 428–436.
mla: Fulek, Radoslav, et al. “Clustered Planarity Testing Revisited.” International
Symposium on Graph Drawing, vol. 8871, Springer Nature, 2014, pp. 428–36,
doi:10.1007/978-3-662-45803-7_36.
short: R. Fulek, J. Kynčl, I. Malinović, D. Pálvölgyi, in:, International Symposium
on Graph Drawing, Springer Nature, Cham, 2014, pp. 428–436.
date_created: 2022-02-25T10:32:14Z
date_published: 2014-01-01T00:00:00Z
date_updated: 2023-02-23T10:08:04Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/978-3-662-45803-7_36
external_id:
arxiv:
- '1305.4519'
intvolume: ' 8871'
language:
- iso: eng
month: '01'
oa_version: Preprint
page: 428-436
place: Cham
publication: International Symposium on Graph Drawing
publication_identifier:
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
related_material:
record:
- id: '1642'
relation: later_version
status: public
scopus_import: '1'
status: public
title: Clustered planarity testing revisited
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 8871
year: '2014'
...
---
_id: '1842'
abstract:
- lang: eng
text: We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2
outerplanar triangulations in both convex and general cases. We also prove that
the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by
O(n3) and O(n10), in the convex and general case, respectively. We then apply
similar methods to prove an (Formula presented.) upper bound on the Ramsey number
of a path with n ordered vertices.
acknowledgement: Marek Krčál was supported by the ERC Advanced Grant No. 267165.
author:
- first_name: Josef
full_name: Cibulka, Josef
last_name: Cibulka
- first_name: Pu
full_name: Gao, Pu
last_name: Gao
- first_name: Marek
full_name: Krcál, Marek
id: 33E21118-F248-11E8-B48F-1D18A9856A87
last_name: Krcál
- first_name: Tomáš
full_name: Valla, Tomáš
last_name: Valla
- first_name: Pavel
full_name: Valtr, Pavel
last_name: Valtr
citation:
ama: Cibulka J, Gao P, Krcál M, Valla T, Valtr P. On the geometric ramsey number
of outerplanar graphs. Discrete & Computational Geometry. 2014;53(1):64-79.
doi:10.1007/s00454-014-9646-x
apa: Cibulka, J., Gao, P., Krcál, M., Valla, T., & Valtr, P. (2014). On the
geometric ramsey number of outerplanar graphs. Discrete & Computational
Geometry. Springer. https://doi.org/10.1007/s00454-014-9646-x
chicago: Cibulka, Josef, Pu Gao, Marek Krcál, Tomáš Valla, and Pavel Valtr. “On
the Geometric Ramsey Number of Outerplanar Graphs.” Discrete & Computational
Geometry. Springer, 2014. https://doi.org/10.1007/s00454-014-9646-x.
ieee: J. Cibulka, P. Gao, M. Krcál, T. Valla, and P. Valtr, “On the geometric ramsey
number of outerplanar graphs,” Discrete & Computational Geometry, vol.
53, no. 1. Springer, pp. 64–79, 2014.
ista: Cibulka J, Gao P, Krcál M, Valla T, Valtr P. 2014. On the geometric ramsey
number of outerplanar graphs. Discrete & Computational Geometry. 53(1), 64–79.
mla: Cibulka, Josef, et al. “On the Geometric Ramsey Number of Outerplanar Graphs.”
Discrete & Computational Geometry, vol. 53, no. 1, Springer, 2014,
pp. 64–79, doi:10.1007/s00454-014-9646-x.
short: J. Cibulka, P. Gao, M. Krcál, T. Valla, P. Valtr, Discrete & Computational
Geometry 53 (2014) 64–79.
date_created: 2018-12-11T11:54:18Z
date_published: 2014-11-14T00:00:00Z
date_updated: 2021-01-12T06:53:33Z
day: '14'
department:
- _id: UlWa
- _id: HeEd
doi: 10.1007/s00454-014-9646-x
intvolume: ' 53'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1310.7004
month: '11'
oa: 1
oa_version: Submitted Version
page: 64 - 79
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '5260'
scopus_import: 1
status: public
title: On the geometric ramsey number of outerplanar graphs
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 53
year: '2014'
...
---
_id: '2154'
abstract:
- lang: eng
text: A result of Boros and Füredi (d = 2) and of Bárány (arbitrary d) asserts that
for every d there exists cd > 0 such that for every n-point set P ⊂ ℝd, some
point of ℝd is covered by at least (Formula presented.) of the d-simplices spanned
by the points of P. The largest possible value of cd has been the subject of ongoing
research. Recently Gromov improved the existing lower bounds considerably by introducing
a new, topological proof method. We provide an exposition of the combinatorial
component of Gromov's approach, in terms accessible to combinatorialists and discrete
geometers, and we investigate the limits of his method. In particular, we give
tighter bounds on the cofilling profiles for the (n - 1)-simplex. These bounds
yield a minor improvement over Gromov's lower bounds on cd for large d, but they
also show that the room for further improvement through the cofilling profiles
alone is quite small. We also prove a slightly better lower bound for c3 by an
approach using an additional structure besides the cofilling profiles. We formulate
a combinatorial extremal problem whose solution might perhaps lead to a tight
lower bound for cd.
acknowledgement: Swiss National Science Foundation (SNF 200021-125309, 200020-138230,
200020-12507)
author:
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: Matoušek J, Wagner U. On Gromov’s method of selecting heavily covered points.
Discrete & Computational Geometry. 2014;52(1):1-33. doi:10.1007/s00454-014-9584-7
apa: Matoušek, J., & Wagner, U. (2014). On Gromov’s method of selecting heavily
covered points. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/s00454-014-9584-7
chicago: Matoušek, Jiří, and Uli Wagner. “On Gromov’s Method of Selecting Heavily
Covered Points.” Discrete & Computational Geometry. Springer, 2014.
https://doi.org/10.1007/s00454-014-9584-7.
ieee: J. Matoušek and U. Wagner, “On Gromov’s method of selecting heavily covered
points,” Discrete & Computational Geometry, vol. 52, no. 1. Springer,
pp. 1–33, 2014.
ista: Matoušek J, Wagner U. 2014. On Gromov’s method of selecting heavily covered
points. Discrete & Computational Geometry. 52(1), 1–33.
mla: Matoušek, Jiří, and Uli Wagner. “On Gromov’s Method of Selecting Heavily Covered
Points.” Discrete & Computational Geometry, vol. 52, no. 1, Springer,
2014, pp. 1–33, doi:10.1007/s00454-014-9584-7.
short: J. Matoušek, U. Wagner, Discrete & Computational Geometry 52 (2014) 1–33.
date_created: 2018-12-11T11:56:01Z
date_published: 2014-07-01T00:00:00Z
date_updated: 2021-01-12T06:55:38Z
day: '01'
department:
- _id: UlWa
doi: 10.1007/s00454-014-9584-7
intvolume: ' 52'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1102.3515
month: '07'
oa: 1
oa_version: Submitted Version
page: 1 - 33
project:
- _id: 25FA3206-B435-11E9-9278-68D0E5697425
grant_number: PP00P2_138948
name: 'Embeddings in Higher Dimensions: Algorithms and Combinatorics'
publication: Discrete & Computational Geometry
publication_status: published
publisher: Springer
publist_id: '4852'
quality_controlled: '1'
scopus_import: 1
status: public
title: On Gromov's method of selecting heavily covered points
type: journal_article
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
volume: 52
year: '2014'
...
---
_id: '2157'
abstract:
- lang: eng
text: 'We show that the following algorithmic problem is decidable: given a 2-dimensional
simplicial complex, can it be embedded (topologically, or equivalently, piecewise
linearly) in ℝ3? By a known reduction, it suffices to decide the embeddability
of a given triangulated 3-manifold X into the 3-sphere S3. The main step, which
allows us to simplify X and recurse, is in proving that if X can be embedded in
S3, then there is also an embedding in which X has a short meridian, i.e., an
essential curve in the boundary of X bounding a disk in S3 nX with length bounded
by a computable function of the number of tetrahedra of X.'
acknowledgement: ERC Advanced Grant No. 267165; Grant GRADR Eurogiga GIG/11/E023 (SNSF-PP00P2-138948);
Swiss National Science Foundation (SNSF-200020-138230).
author:
- first_name: Jiří
full_name: Matoušek, Jiří
last_name: Matoušek
- first_name: Eric
full_name: Sedgwick, Eric
last_name: Sedgwick
- first_name: Martin
full_name: Tancer, Martin
id: 38AC689C-F248-11E8-B48F-1D18A9856A87
last_name: Tancer
orcid: 0000-0002-1191-6714
- first_name: Uli
full_name: Wagner, Uli
id: 36690CA2-F248-11E8-B48F-1D18A9856A87
last_name: Wagner
orcid: 0000-0002-1494-0568
citation:
ama: 'Matoušek J, Sedgwick E, Tancer M, Wagner U. Embeddability in the 3 sphere
is decidable. In: Proceedings of the Annual Symposium on Computational Geometry.
ACM; 2014:78-84. doi:10.1145/2582112.2582137'
apa: 'Matoušek, J., Sedgwick, E., Tancer, M., & Wagner, U. (2014). Embeddability
in the 3 sphere is decidable. In Proceedings of the Annual Symposium on Computational
Geometry (pp. 78–84). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582137'
chicago: Matoušek, Jiří, Eric Sedgwick, Martin Tancer, and Uli Wagner. “Embeddability
in the 3 Sphere Is Decidable.” In Proceedings of the Annual Symposium on Computational
Geometry, 78–84. ACM, 2014. https://doi.org/10.1145/2582112.2582137.
ieee: J. Matoušek, E. Sedgwick, M. Tancer, and U. Wagner, “Embeddability in the
3 sphere is decidable,” in Proceedings of the Annual Symposium on Computational
Geometry, Kyoto, Japan, 2014, pp. 78–84.
ista: 'Matoušek J, Sedgwick E, Tancer M, Wagner U. 2014. Embeddability in the 3
sphere is decidable. Proceedings of the Annual Symposium on Computational Geometry.
SoCG: Symposium on Computational Geometry, 78–84.'
mla: Matoušek, Jiří, et al. “Embeddability in the 3 Sphere Is Decidable.” Proceedings
of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 78–84, doi:10.1145/2582112.2582137.
short: J. Matoušek, E. Sedgwick, M. Tancer, U. Wagner, in:, Proceedings of the Annual
Symposium on Computational Geometry, ACM, 2014, pp. 78–84.
conference:
end_date: 2014-06-11
location: Kyoto, Japan
name: 'SoCG: Symposium on Computational Geometry'
start_date: 2014-06-08
date_created: 2018-12-11T11:56:02Z
date_published: 2014-06-01T00:00:00Z
date_updated: 2023-09-11T13:38:49Z
day: '01'
department:
- _id: UlWa
doi: 10.1145/2582112.2582137
language:
- iso: eng
main_file_link:
- open_access: '1'
url: http://arxiv.org/abs/1402.0815
month: '06'
oa: 1
oa_version: Submitted Version
page: 78 - 84
publication: Proceedings of the Annual Symposium on Computational Geometry
publication_status: published
publisher: ACM
publist_id: '4849'
quality_controlled: '1'
related_material:
record:
- id: '425'
relation: later_version
status: public
scopus_import: 1
status: public
title: Embeddability in the 3 sphere is decidable
type: conference
user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87
year: '2014'
...