---
_id: '15082'
abstract:
- lang: eng
  text: "Two plane drawings of geometric graphs on the same set of points are called
    disjoint compatible if their union is plane and they do not have an edge in common.
    For a given set S of 2n points two plane drawings of perfect matchings M1 and
    M2 (which do not need to be disjoint nor compatible) are disjoint tree-compatible
    if there exists a plane drawing of a spanning tree T on S which is disjoint compatible
    to both M1 and M2.\r\nWe show that the graph of all disjoint tree-compatible perfect
    geometric matchings on 2n points in convex position is connected if and only if
    2n ≥ 10. Moreover, in that case the diameter\r\nof this graph is either 4 or 5,
    independent of n."
acknowledgement: Research on this work was initiated at the 6th Austrian-Japanese-Mexican-Spanish
  Workshop on Discrete Geometry and continued during the 16th European Geometric Graph-Week,
  both held near Strobl, Austria. We are grateful to the participants for the inspiring
  atmosphere. We especially thank Alexander Pilz for bringing this class of problems
  to our attention and Birgit Vogtenhuber for inspiring discussions. D.P. is partially
  supported by the FWF grant I 3340-N35 (Collaborative DACH project Arrangements and
  Drawings). The research stay of P.P. at IST Austria is funded by the project CZ.02.2.69/0.0/0.0/17_050/0008466
  Improvement of internationalization in the field of research and development at
  Charles University, through the support of quality projects MSCA-IF. This project
  has received funding from the European Union’s Horizon 2020 research and innovation
  programme under the Marie Skłodowska-Curie grant agreement No 734922.
article_number: '56'
article_processing_charge: No
author:
- first_name: Oswin
  full_name: Aichholzer, Oswin
  last_name: Aichholzer
- first_name: Julia
  full_name: Obmann, Julia
  last_name: Obmann
- first_name: Pavel
  full_name: Patak, Pavel
  id: B593B804-1035-11EA-B4F1-947645A5BB83
  last_name: Patak
- first_name: Daniel
  full_name: Perz, Daniel
  last_name: Perz
- first_name: Josef
  full_name: Tkadlec, Josef
  id: 3F24CCC8-F248-11E8-B48F-1D18A9856A87
  last_name: Tkadlec
  orcid: 0000-0002-1097-9684
citation:
  ama: 'Aichholzer O, Obmann J, Patak P, Perz D, Tkadlec J. Disjoint tree-compatible
    plane perfect matchings. In: <i>36th European Workshop on Computational Geometry</i>.
    ; 2020.'
  apa: Aichholzer, O., Obmann, J., Patak, P., Perz, D., &#38; Tkadlec, J. (2020).
    Disjoint tree-compatible plane perfect matchings. In <i>36th European Workshop
    on Computational Geometry</i>. Würzburg, Germany, Virtual.
  chicago: Aichholzer, Oswin, Julia Obmann, Pavel Patak, Daniel Perz, and Josef Tkadlec.
    “Disjoint Tree-Compatible Plane Perfect Matchings.” In <i>36th European Workshop
    on Computational Geometry</i>, 2020.
  ieee: O. Aichholzer, J. Obmann, P. Patak, D. Perz, and J. Tkadlec, “Disjoint tree-compatible
    plane perfect matchings,” in <i>36th European Workshop on Computational Geometry</i>,
    Würzburg, Germany, Virtual, 2020.
  ista: 'Aichholzer O, Obmann J, Patak P, Perz D, Tkadlec J. 2020. Disjoint tree-compatible
    plane perfect matchings. 36th European Workshop on Computational Geometry. EuroCG:
    European Workshop on Computational Geometry, 56.'
  mla: Aichholzer, Oswin, et al. “Disjoint Tree-Compatible Plane Perfect Matchings.”
    <i>36th European Workshop on Computational Geometry</i>, 56, 2020.
  short: O. Aichholzer, J. Obmann, P. Patak, D. Perz, J. Tkadlec, in:, 36th European
    Workshop on Computational Geometry, 2020.
conference:
  end_date: 2020-03-18
  location: Würzburg, Germany, Virtual
  name: 'EuroCG: European Workshop on Computational Geometry'
  start_date: 2020-03-16
date_created: 2024-03-05T08:57:17Z
date_published: 2020-04-01T00:00:00Z
date_updated: 2024-03-05T09:00:07Z
day: '01'
department:
- _id: KrCh
- _id: UlWa
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://www1.pub.informatik.uni-wuerzburg.de/eurocg2020/data/uploads/papers/eurocg20_paper_56.pdf
month: '04'
oa: 1
oa_version: Published Version
publication: 36th European Workshop on Computational Geometry
publication_status: published
quality_controlled: '1'
status: public
title: Disjoint tree-compatible plane perfect matchings
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2020'
...
---
_id: '7108'
abstract:
- lang: eng
  text: We prove that for every d ≥ 2, deciding if a pure, d-dimensional, simplicial
    complex is shellable is NP-hard, hence NP-complete. This resolves a question raised,
    e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d
    ≥ 2 and k ≥ 0, deciding if a pure, d-dimensional, simplicial complex is k-decomposable
    is NP-hard. For d ≥ 3, both problems remain NP-hard when restricted to contractible
    pure d-dimensional complexes. Another simple corollary of our result is that it
    is NP-hard to decide whether a given poset is CL-shellable.
article_number: '21'
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Xavier
  full_name: Goaoc, Xavier
  last_name: Goaoc
- first_name: Pavel
  full_name: Patak, Pavel
  id: B593B804-1035-11EA-B4F1-947645A5BB83
  last_name: Patak
- first_name: Zuzana
  full_name: Patakova, Zuzana
  id: 48B57058-F248-11E8-B48F-1D18A9856A87
  last_name: Patakova
  orcid: 0000-0002-3975-1683
- first_name: Martin
  full_name: Tancer, Martin
  last_name: Tancer
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
citation:
  ama: Goaoc X, Patak P, Patakova Z, Tancer M, Wagner U. Shellability is NP-complete.
    <i>Journal of the ACM</i>. 2019;66(3). doi:<a href="https://doi.org/10.1145/3314024">10.1145/3314024</a>
  apa: Goaoc, X., Patak, P., Patakova, Z., Tancer, M., &#38; Wagner, U. (2019). Shellability
    is NP-complete. <i>Journal of the ACM</i>. ACM. <a href="https://doi.org/10.1145/3314024">https://doi.org/10.1145/3314024</a>
  chicago: Goaoc, Xavier, Pavel Patak, Zuzana Patakova, Martin Tancer, and Uli Wagner.
    “Shellability Is NP-Complete.” <i>Journal of the ACM</i>. ACM, 2019. <a href="https://doi.org/10.1145/3314024">https://doi.org/10.1145/3314024</a>.
  ieee: X. Goaoc, P. Patak, Z. Patakova, M. Tancer, and U. Wagner, “Shellability is
    NP-complete,” <i>Journal of the ACM</i>, vol. 66, no. 3. ACM, 2019.
  ista: Goaoc X, Patak P, Patakova Z, Tancer M, Wagner U. 2019. Shellability is NP-complete.
    Journal of the ACM. 66(3), 21.
  mla: Goaoc, Xavier, et al. “Shellability Is NP-Complete.” <i>Journal of the ACM</i>,
    vol. 66, no. 3, 21, ACM, 2019, doi:<a href="https://doi.org/10.1145/3314024">10.1145/3314024</a>.
  short: X. Goaoc, P. Patak, Z. Patakova, M. Tancer, U. Wagner, Journal of the ACM
    66 (2019).
date_created: 2019-11-26T10:13:59Z
date_published: 2019-06-01T00:00:00Z
date_updated: 2023-09-06T11:10:58Z
day: '01'
department:
- _id: UlWa
doi: 10.1145/3314024
external_id:
  arxiv:
  - '1711.08436'
  isi:
  - '000495406300007'
intvolume: '        66'
isi: 1
issue: '3'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/pdf/1711.08436.pdf
month: '06'
oa: 1
oa_version: Preprint
publication: Journal of the ACM
publication_identifier:
  issn:
  - 0004-5411
publication_status: published
publisher: ACM
quality_controlled: '1'
related_material:
  record:
  - id: '184'
    relation: earlier_version
    status: public
scopus_import: '1'
status: public
title: Shellability is NP-complete
type: journal_article
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
volume: 66
year: '2019'
...