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Matching Edges and Faces in Polygonal Partitions O. Aichholzer 1 F. Aurenhammer 2 P. GonzalezNava 3 T. Hackl 4

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.
1 Introduction
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 [2] 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


Source: Aurenhammer, Franz - Institute for Theoretical Computer Science, Technische Universität Graz


Collections: Computer Technologies and Information Sciences