Department of
Applied
Mathematics

Connectivity of Triangulation Flip Graphs in the Plane

Emo Welzl

ETH Zürich

Abstract

In a straight line embedded triangulation of a point set P in the plane, removing an inner edge and - provided the resulting quadrilateral Q is convex - adding the other diagonal is called an edge flip. The 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. This presentation is towards a better understanding of this graph, with emphasis on its connectivity.

It is known that every triangulation allows at least n/2-2 edge flips and we show (n/2-2)-vertex connectivity for flip graphs of all P in general position, n:=|P|. Somewhat stronger, but restricted to P large enough, we show that the vertex connectivity is determined by the minimum degree occurring in the flip graph, i.e. the minimum number of flippable edges in any triangulation of P.

A corresponding result is shown for so-called partial triangulations, i.e. the set of all triangulations of subsets of P which contain all extreme points of P. Here the flip operation is extended to bistellar flips (edge flip, and insertion and removal of an inner vertex of degree three). We prove (n-3)-edge connectedness for all P in general position and (n-3)-vertex connectedness for n large enough ((n-3) is tight, since there is always a partial triangulation which allows exactly n-3 bistellar flips). This matches the situation known (through the secondary polytope) for regular triangulations (i.e. partial triangulations obtained by lifting the points and projecting the lower convex hull).

This is joint work with Uli Wagner, IST Austria.