99 research outputs found
The polytope of non-crossing graphs on a planar point set
For any finite set \A of points in , we define a
-dimensional simple polyhedron whose face poset is isomorphic to the
poset of ``non-crossing marked graphs'' with vertex set \A, where a marked
graph is defined as a geometric graph together with a subset of its vertices.
The poset of non-crossing graphs on \A appears as the complement of the star
of a face in that polyhedron.
The polyhedron has a unique maximal bounded face, of dimension
where is the number of points of \A in the interior of \conv(\A). The
vertices of this polytope are all the pseudo-triangulations of \A, and the
edges are flips of two types: the traditional diagonal flips (in
pseudo-triangulations) and the removal or insertion of a single edge.
As a by-product of our construction we prove that all pseudo-triangulations
are infinitesimally rigid graphs.Comment: 28 pages, 16 figures. Main change from v1 and v2: Introduction has
been reshape
On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions
Let be a collection of points in the plane, each moving along some
straight line at unit speed. We obtain an almost tight upper bound of
, for any , on the maximum number of discrete
changes that the Delaunay triangulation of experiences
during this motion. Our analysis is cast in a purely topological setting, where
we only assume that (i) any four points can be co-circular at most three times,
and (ii) no triple of points can be collinear more than twice; these
assumptions hold for unit speed motions.Comment: 138 pages+ Appendix of 7 pages. A preliminary version has appeared in
Proceedings of the 54th Annual Symposium on Foundations of Computer Science
(FOCS 2013). The paper extends the result of http://arxiv.org/abs/1304.3671
to more general motions. The presentation is self-contained with main ideas
delivered in Sections 1--
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