 
Summary: Convexity Minimizes PseudoTriangulations
Oswin Aichholzer Franz Aurenhammer y Hannes Krasser yz Bettina Speckmann x
Abstract
The number of minimum pseudotriangulations is minimized for point sets in convex position.
1 Introduction
A pseudotriangle is a planar polygon with exactly three convex vertices, called corners. Three
re
ex chains of edges join the corners. Let S be a set of n points in general position in the plane. A
pseudotriangulation for S is a partition of the convex hull of S into pseudotriangles whose vertex
set is S. A pseudotriangulation is called minimum if it consists of exactly n 2 pseudotriangles
(and 2n 3 edges), the minimum possible. Each vertex of a minimum pseudotriangulation is
pointed, that is, its incident edges span a convex angle. In fact, minimum pseudotriangulations
can be characterized as maximal planar straightline graphs where each vertex is pointed [18].
Therefore, they have been alternatively called pointed pseudotriangulations.
Pseudotriangulations have received considerable attention in computational geometry due to
their applications to visibility [13, 14], ray shooting [8], kinetic collision detection [1, 11, 12], rigid
ity [18], and guarding [17]. Several of their interesting geometric and combinatorial properties
have been discovered recently [16, 10, 2, 9]. Still, little is known about the number of pseudo
triangulations a general point set S allows. (Assuming general position of S is necessary to avoid
trivial situations.) In [15], the number of minimum pseudotriangulations is determined for sets of
points with exactly one interior point. Also, a (coarse) upper bound on the number of minimum
