818 research outputs found
Embedding Stacked Polytopes on a Polynomial-Size Grid
A stacking operation adds a -simplex on top of a facet of a simplicial
-polytope while maintaining the convexity of the polytope. A stacked
-polytope is a polytope that is obtained from a -simplex and a series of
stacking operations. We show that for a fixed every stacked -polytope
with vertices can be realized with nonnegative integer coordinates. The
coordinates are bounded by , except for one axis, where the
coordinates are bounded by . The described realization can be
computed with an easy algorithm.
The realization of the polytopes is obtained with a lifting technique which
produces an embedding on a large grid. We establish a rounding scheme that
places the vertices on a sparser grid, while maintaining the convexity of the
embedding.Comment: 22 pages, 10 Figure
09451 Abstracts Collection -- Geometric Networks, Metric Space Embeddings and Spatial Data Mining
From November 1 to 6, 2009, the Dagstuhl Seminar 09451 ``Geometric Networks, Metric Space Embeddings and Spatial Data Mining\u27\u27 was held
in Schloss Dagstuhl~--~Leibniz Center for Informatics.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Planar projections of graphs
We introduce and study a new graph representation where vertices are embedded
in three or more dimensions, and in which the edges are drawn on the
projections onto the axis-parallel planes. We show that the complete graph on
vertices has a representation in planes. In 3
dimensions, we show that there exist graphs with edges that can be
projected onto two orthogonal planes, and that this is best possible. Finally,
we obtain bounds in terms of parameters such as geometric thickness and linear
arboricity. Using such a bound, we show that every graph of maximum degree 5
has a plane-projectable representation in 3 dimensions.Comment: Accepted at CALDAM 202
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