---
_id: '7807'
abstract:
- lang: eng
  text: "In a straight-line embedded triangulation of a point set P in the plane,
    removing an inner edge and—provided the resulting quadrilateral is convex—adding
    the other diagonal is called an edge flip. The (edge) flip graph has all triangulations
    as vertices, and a pair of triangulations is adjacent if they can be obtained
    from each other by an edge flip. The goal of this paper is to contribute to a
    better understanding of the flip graph, with an emphasis on its connectivity.\r\nFor
    sets in general position, it is known that every triangulation allows at least
    edge flips (a tight bound) which gives the minimum degree of any flip graph for
    n points. We show that for every point set P in general position, the flip graph
    is at least -vertex connected. Somewhat more strongly, we show that the vertex
    connectivity equals the minimum degree occurring in the flip graph, i.e. the minimum
    number of flippable edges in any triangulation of P, provided P is large enough.
    Finally, we exhibit some of the geometry of the flip graph by showing that the
    flip graph can be covered by 1-skeletons of polytopes of dimension (products of
    associahedra).\r\nA corresponding result ((n – 3)-vertex connectedness) can be
    shown for the bistellar flip graph of partial triangulations, i.e. the set of
    all triangulations of subsets of P which contain all extreme points of P. This
    will be treated separately in a second part."
article_processing_charge: No
arxiv: 1
author:
- first_name: Uli
  full_name: Wagner, Uli
  id: 36690CA2-F248-11E8-B48F-1D18A9856A87
  last_name: Wagner
  orcid: 0000-0002-1494-0568
- first_name: Emo
  full_name: Welzl, Emo
  last_name: Welzl
citation:
  ama: 'Wagner U, Welzl E. Connectivity of triangulation flip graphs in the plane
    (Part I: Edge flips). In: <i>Proceedings of the Annual ACM-SIAM Symposium on Discrete
    Algorithms</i>. Vol 2020-January. SIAM; 2020:2823-2841. doi:<a href="https://doi.org/10.1137/1.9781611975994.172">10.1137/1.9781611975994.172</a>'
  apa: 'Wagner, U., &#38; Welzl, E. (2020). Connectivity of triangulation flip graphs
    in the plane (Part I: Edge flips). In <i>Proceedings of the Annual ACM-SIAM Symposium
    on Discrete Algorithms</i> (Vol. 2020–January, pp. 2823–2841). Salt Lake City,
    UT, United States: SIAM. <a href="https://doi.org/10.1137/1.9781611975994.172">https://doi.org/10.1137/1.9781611975994.172</a>'
  chicago: 'Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs
    in the Plane (Part I: Edge Flips).” In <i>Proceedings of the Annual ACM-SIAM Symposium
    on Discrete Algorithms</i>, 2020–January:2823–41. SIAM, 2020. <a href="https://doi.org/10.1137/1.9781611975994.172">https://doi.org/10.1137/1.9781611975994.172</a>.'
  ieee: 'U. Wagner and E. Welzl, “Connectivity of triangulation flip graphs in the
    plane (Part I: Edge flips),” in <i>Proceedings of the Annual ACM-SIAM Symposium
    on Discrete Algorithms</i>, Salt Lake City, UT, United States, 2020, vol. 2020–January,
    pp. 2823–2841.'
  ista: 'Wagner U, Welzl E. 2020. Connectivity of triangulation flip graphs in the
    plane (Part I: Edge flips). Proceedings of the Annual ACM-SIAM Symposium on Discrete
    Algorithms. SODA: Symposium on Discrete Algorithms vol. 2020–January, 2823–2841.'
  mla: 'Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs in
    the Plane (Part I: Edge Flips).” <i>Proceedings of the Annual ACM-SIAM Symposium
    on Discrete Algorithms</i>, vol. 2020–January, SIAM, 2020, pp. 2823–41, doi:<a
    href="https://doi.org/10.1137/1.9781611975994.172">10.1137/1.9781611975994.172</a>.'
  short: U. Wagner, E. Welzl, in:, Proceedings of the Annual ACM-SIAM Symposium on
    Discrete Algorithms, SIAM, 2020, pp. 2823–2841.
conference:
  end_date: 2020-01-08
  location: Salt Lake City, UT, United States
  name: 'SODA: Symposium on Discrete Algorithms'
  start_date: 2020-01-05
date_created: 2020-05-10T22:00:48Z
date_published: 2020-01-01T00:00:00Z
date_updated: 2024-10-09T21:03:33Z
day: '01'
department:
- _id: UlWa
doi: 10.1137/1.9781611975994.172
external_id:
  arxiv:
  - '2003.13557'
fulldoi: https://doi.org/10.1137/1.9781611975994.172
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1137/1.9781611975994.172
month: '01'
oa: 1
oa_version: Submitted Version
page: 2823-2841
publication: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
publication_identifier:
  isbn:
  - '9781611975994'
publication_status: published
publisher: SIAM
quality_controlled: '1'
related_material:
  record:
  - id: '12129'
    relation: later_version
    status: public
scopus_import: 1
status: public
title: 'Connectivity of triangulation flip graphs in the plane (Part I: Edge flips)'
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 2020-January
year: '2020'
...