Connectivity of Triangulation Flip Graphs in the Plane

Abstract

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 $$P'$$ 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 $$P \setminus P'$$ 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 $$\lceil {n}/{2}-2\rceil $$ ⌈ n / 2 - 2 ⌉ -vertex connected, and the bistellar flip graph is $$(n-3)$$ ( 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 $$(n-3)$$ ( 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 $$\lceil {n}/{2} -2\rceil $$ ⌈ n / 2 - 2 ⌉ (products of associahedra) and the bistellar flip graph can be covered by graphs of polytopes of dimension $$n-3$$ n - 3 (products of secondary polytopes). (ii) A partial triangulation is regular, if it has distance $$n-3$$ 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 $$n-3$$ 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.