On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph

Abstract: This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega be a height function which lifts the vertices of A into R3. Every flip in triangulations of A can be associated with a direction. We first established a relatively obvious relation between monotone sequences of directed flips between triangulations of A and triangulations of the lifted point set of A in R3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case if this result in 3d has been shown by B.Joe in 1989. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra. We proved sufficient conditions for the termination of this algorithm and show that it runs in O(n3) time.