article
Connectivity of triangulation flip graphs in the plane
published
yes
Uli
Wagner
author 36690CA2-F248-11E8-B48F-1D18A9856A870000-0002-1494-0568
Emo
Welzl
author
UlWa
department
Given a finite point set P in general position in the plane, a full triangulation of P is a maximal straight-line embedded plane graph on P. A partial triangulation of P is a full triangulation of some subset P′ of P containing all extreme points in P. A bistellar flip on a partial triangulation either flips an edge (called edge flip), removes a non-extreme point of degree 3, or adds a point in P∖P′ as vertex of degree 3. The bistellar flip graph has all partial triangulations as vertices, and a pair of partial triangulations is adjacent if they can be obtained from one another by a bistellar flip. The edge flip graph is defined with full triangulations as vertices, and edge flips determining the adjacencies. Lawson showed in the early seventies that these graphs are connected. The goal of this paper is to investigate the structure of these graphs, with emphasis on their vertex connectivity. For sets P of n points in the plane in general position, we show that the edge flip graph is ⌈n/2−2⌉-vertex connected, and the bistellar flip graph is (n−3)-vertex connected; both results are tight. The latter bound matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points to 3-space and projecting back the lower convex hull), where (n−3)-vertex connectivity has been known since the late eighties through the secondary polytope due to Gelfand, Kapranov, & Zelevinsky and Balinski’s Theorem. For the edge flip-graph, we additionally show that the vertex connectivity is at least as large as (and hence equal to) the minimum degree (i.e., the minimum number of flippable edges in any full triangulation), provided that n is large enough. Our methods also yield several other results: (i) The edge flip graph can be covered by graphs of polytopes of dimension ⌈n/2−2⌉ (products of associahedra) and the bistellar flip graph can be covered by graphs of polytopes of dimension n−3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance n−3 in the Hasse diagram of the partial order of partial subdivisions from the trivial subdivision. (iii) All partial triangulations of a point set are regular iff the partial order of partial subdivisions has height n−3. (iv) There are arbitrarily large sets P with non-regular partial triangulations and such that every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular triangulations.
https://research-explorer.ista.ac.at/download/12129/12345/2022_DiscreteCompGeometry_Wagner.pdf
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https://creativecommons.org/licenses/by/4.0/
Springer Nature2022
eng
Computational Theory and MathematicsDiscrete Mathematics and CombinatoricsGeometry and TopologyTheoretical Computer Science
Discrete & Computational Geometry
0179-5376
1432-0444
00088322220000310.1007/s00454-022-00436-2
6841227-1284
https://research-explorer.ista.ac.at/record/7807 https://research-explorer.ista.ac.at/record/7990
Wagner, U., & Welzl, E. (2022). Connectivity of triangulation flip graphs in the plane. <i>Discrete & Computational Geometry</i>. Springer Nature. <a href="https://doi.org/10.1007/s00454-022-00436-2">https://doi.org/10.1007/s00454-022-00436-2</a>
Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs in the Plane.” <i>Discrete & Computational Geometry</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s00454-022-00436-2">https://doi.org/10.1007/s00454-022-00436-2</a>.
U. Wagner and E. Welzl, “Connectivity of triangulation flip graphs in the plane,” <i>Discrete & Computational Geometry</i>, vol. 68, no. 4. Springer Nature, pp. 1227–1284, 2022.
Wagner U, Welzl E. Connectivity of triangulation flip graphs in the plane. <i>Discrete & Computational Geometry</i>. 2022;68(4):1227-1284. doi:<a href="https://doi.org/10.1007/s00454-022-00436-2">10.1007/s00454-022-00436-2</a>
Wagner, Uli, and Emo Welzl. “Connectivity of Triangulation Flip Graphs in the Plane.” <i>Discrete & Computational Geometry</i>, vol. 68, no. 4, Springer Nature, 2022, pp. 1227–84, doi:<a href="https://doi.org/10.1007/s00454-022-00436-2">10.1007/s00454-022-00436-2</a>.
Wagner U, Welzl E. 2022. Connectivity of triangulation flip graphs in the plane. Discrete & Computational Geometry. 68(4), 1227–1284.
U. Wagner, E. Welzl, Discrete & Computational Geometry 68 (2022) 1227–1284.
121292023-01-12T12:02:28Z2024-10-09T21:03:33Z