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Summary: Convexity Minimizes Pseudo-Triangulations
Oswin Aichholzer Franz Aurenhammer y Hannes Krasser yz Bettina Speckmann x
Abstract
The number of minimum pseudo-triangulations is minimized for point sets in convex position.
1 Introduction
A pseudo-triangle 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
pseudo-triangulation for S is a partition of the convex hull of S into pseudo-triangles whose vertex
set is S. A pseudo-triangulation is called minimum if it consists of exactly n 2 pseudo-triangles
(and 2n 3 edges), the minimum possible. Each vertex of a minimum pseudo-triangulation is
pointed, that is, its incident edges span a convex angle. In fact, minimum pseudo-triangulations
can be characterized as maximal planar straight-line graphs where each vertex is pointed [18].
Therefore, they have been alternatively called pointed pseudo-triangulations.
Pseudo-triangulations 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 pseudo-triangulations is determined for sets of
points with exactly one interior point. Also, a (coarse) upper bound on the number of minimum
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