Connectivity of Triangulation Flip Graphs in the Plane (Part II: Bistellar Flips)

Authors Uli Wagner , Emo Welzl

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Author Details

Uli Wagner
  • IST Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Emo Welzl
  • Department of Computer Science, ETH Zürich, Switzerland


This research started at the 11th Gremo’s Workshop on Open Problems (GWOP), Alp Sellamatt, Switzerland, June 24-28, 2013, motivated by a question posed by Filip Morić. We thank Michael Joswig, Jesús De Loera, and Francisco Santos for helpful discussions on the topics of this paper, and Daniel Bertschinger for carefully reading an earlier version and for many helpful comments.

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Uli Wagner and Emo Welzl. Connectivity of Triangulation Flip Graphs in the Plane (Part II: Bistellar Flips). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 67:1-67:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


Given a finite point set P in general position in the plane, a full triangulation is a maximal straight-line embedded plane graph on P. A partial triangulation on 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, 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 goal of this paper is to investigate the structure of this graph, with emphasis on its connectivity. For sets P of n points in general position, we show that the bistellar flip graph is (n-3)-connected, thereby answering, for sets in general position, an open questions raised in a book (by De Loera, Rambau, and Santos) and a survey (by Lee and Santos) on triangulations. This matches the situation for the subfamily of regular triangulations (i.e., partial triangulations obtained by lifting the points and projecting the lower convex hull), where (n-3)-connectivity has been known since the late 1980s through the secondary polytope (Gelfand, Kapranov, Zelevinsky) and Balinski’s Theorem. Our methods also yield the following results (see the full version [Wagner and Welzl, 2020]): (i) 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 are regular iff the trivial subdivision has height n-3 in the partial order of partial subdivisions. (iv) There are arbitrarily large sets P with non-regular partial triangulations, while every proper subset has only regular triangulations, i.e., there are no small certificates for the existence of non-regular partial triangulations (answering a question by F. Santos in the unexpected direction).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • triangulation
  • flip graph
  • graph connectivity
  • associahedron
  • subdivision
  • convex decomposition
  • flippable edge
  • flip complex
  • regular triangulation
  • bistellar flip graph
  • secondary polytope
  • polyhedral subdivision


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