 
Summary: Adapting (Pseudo)Triangulations with
a NearLinear Number of Edge Flips
Oswin Aichholzer 1 , Franz Aurenhammer, Hannes Krasser 2
1 Introduction
In geometric data processing, structures that partition the geometric input, as well as
connectivity structures for geometric objects, play an important role. A versatile tool
in this context are triangular meshes, often called triangulations; see e.g., the survey
articles [6, 12, 5]. A triangulation of a finite set S of points in the plane is a maximal
planar straightline graph that uses all and only the points in S as its vertices. Each
face in a triangulation is a triangle spanned by S.
In the last few years, a relaxation of triangulations, called pseudotriangulations
(or geodesic triangulations), has received considerable attention. Here, faces bounded
by three concave chains, rather than by three line segments, are allowed. The scope
of applications of pseudotriangulations as a geometric data stucture ranges from ray
shooting [10, 14] and visibility [25, 26] to kinetic collision detection [1, 21, 22], rigid
ity [32, 29, 15], and guarding [31]. Still, only very recently, results on the combina
torial properties of pseudotriangulations have been obtained. These include bounds
on the minimal vertex and face degree [20] and on the number of possible pseudo
triangulations [27, 3].
The usefulness of (pseudo)triangulations partially stems from the fact that these
