Summary: Matching Edges and Faces in Polygonal Partitions
O. Aichholzer 1 F. Aurenhammer 2 P. GonzalezNava 3 T. Hackl 4
C. Huemer 5 F. Hurtado 6 H. Krasser 7 S. Ray 8 B. Vogtenhuber 9
We define general Laman (count) conditions for edges and faces of polygonal partitions
in the plane. Several wellknown classes, including kregular partitions, kangulations, and
rankk pseudotriangulations, are shown to fulfill such conditions. As an implication, non
trivial perfect matchings exist between the edge sets (or face sets) of two such structures when
they live on the same point set. We also describe a link to spanning tree decompositions that
applies to quadrangulations and certain pseudotriangulations.
Polygonal partitions in the plane are a versatile tool in computational geometry, and much effort
has gone into the study of their structural properties. In particular, a wealth of results exists for the
class of triangulations of a finite point set in the plane. Less is known for polygonal partitions where
faces are of more general shape, like kangulations, pseudotriangulations, or kregular partitions.
In the present paper, we intend to generalize results of a particular type from triangulations to the
three classes of polygonal partitions mentioned above.
The paper  establishes the existence of certain matchings between two given triangulations
on top of the same point set S. For instance, for any two triangulations T 1 and T 2 of S we can pair
each edge e 1 # T 1 with an edge e 2 # T 2 such that either e 1 = e 2 or e 1 crosses e 2 . Moreover, each