134 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
Embedding Stacked Polytopes on a Polynomial-Size Grid
We show how to realize a stacked 3D polytope (formed
by repeatedly stacking a tetrahedron onto a triangular
face) by a strictly convex embedding with its n vertices
on an integer grid of size O(n4) x O(n4) x O(n18). We
use a perturbation technique to construct an integral 2D
embedding that lifts to a small 3D polytope, all in linear
time. This result solves a question posed by G unter M.
Ziegler, and is the rst nontrivial subexponential upper
bound on the long-standing open question of the
grid size necessary to embed arbitrary convex polyhedra,
that is, about effcient versions of Steinitz's 1916
theorem. An immediate consequence of our result is
that O(log n)-bit coordinates suffice for a greedy routing
strategy in planar 3-trees.Deutsche Forschungsgemeinschaft (DFG) (Grant No. SCHU 2458/1-1
A Quantitative Steinitz Theorem for Plane Triangulations
We give a new proof of Steinitz's classical theorem in the case of plane
triangulations, which allows us to obtain a new general bound on the grid size
of the simplicial polytope realizing a given triangulation, subexponential in a
number of special cases.
Formally, we prove that every plane triangulation with vertices can
be embedded in in such a way that it is the vertical projection
of a convex polyhedral surface. We show that the vertices of this surface may
be placed in a integer grid, where and denotes the shedding diameter of , a
quantity defined in the paper.Comment: 25 pages, 6 postscript figure
Small grid embeddings of 3-polytopes
We introduce an algorithm that embeds a given 3-connected planar graph as a
convex 3-polytope with integer coordinates. The size of the coordinates is
bounded by . If the graph contains a triangle we can
bound the integer coordinates by . If the graph contains a
quadrilateral we can bound the integer coordinates by . The
crucial part of the algorithm is to find a convex plane embedding whose edges
can be weighted such that the sum of the weighted edges, seen as vectors,
cancel at every point. It is well known that this can be guaranteed for the
interior vertices by applying a technique of Tutte. We show how to extend
Tutte's ideas to construct a plane embedding where the weighted vector sums
cancel also on the vertices of the boundary face
A duality transform for realizing convex polytopes with small integer coordinates
Wir entwickeln eine Dualitätstransformation für Polyeder, die eine Einbettung auf dem polynomiellen Gitter berechnet, wenn das ursprüngliche Polyeder auf einem polynomiellen Gitter gegeben ist. Die Konstruktion erfordert einen beschränkten Knotengrad des Polytop-Graphen, funktioniert aber im allgemeinen Fall für die Klasse der Stapelpolytope. Als Konsequenz können wir zeigen, dass sich die "Truncated Polytopes" auf einem polynomiellen Gitter realisieren lassen. Dieses Ergebnis gilt für jede (feste) Dimension.We study realizations of convex polytopes with small integer coordinates. We develop an efficient duality transform, that allows us to go from an efficient realization of a convex polytope to an efficient realization of its dual.Our methods prove to be especially efficient for realizing the class of polytopes dual to stacked polytopes, known as truncated polytopes. We show that every 3d truncated polytope with n vertices can be realized on an integer grid of size O(n^(9lg(6)+1)), and in R^d the required grid size is n^(O(d^2*lg(d))). The class of truncated polytopes is only the second nontrivial class of polytopes, the first being the class of stacked polytopes, for which realizations on a polynomial size integer grid are known to exists
Strongly Monotone Drawings of Planar Graphs
A straight-line drawing of a graph is a monotone drawing if for each pair of
vertices there is a path which is monotonically increasing in some direction,
and it is called a strongly monotone drawing if the direction of monotonicity
is given by the direction of the line segment connecting the two vertices.
We present algorithms to compute crossing-free strongly monotone drawings for
some classes of planar graphs; namely, 3-connected planar graphs, outerplanar
graphs, and 2-trees. The drawings of 3-connected planar graphs are based on
primal-dual circle packings. Our drawings of outerplanar graphs are based on a
new algorithm that constructs strongly monotone drawings of trees which are
also convex. For irreducible trees, these drawings are strictly convex
Adjacency Graphs of Polyhedral Surfaces
We study whether a given graph can be realized as an adjacency graph of the
polygonal cells of a polyhedral surface in . We show that every
graph is realizable as a polyhedral surface with arbitrary polygonal cells, and
that this is not true if we require the cells to be convex. In particular, if
the given graph contains , , or any nonplanar -tree as a
subgraph, no such realization exists. On the other hand, all planar graphs,
, and can be realized with convex cells. The same holds for
any subdivision of any graph where each edge is subdivided at least once, and,
by a result from McMullen et al. (1983), for any hypercube.
Our results have implications on the maximum density of graphs describing
polyhedral surfaces with convex cells: The realizability of hypercubes shows
that the maximum number of edges over all realizable -vertex graphs is in
. From the non-realizability of , we obtain that
any realizable -vertex graph has edges. As such, these graphs
can be considerably denser than planar graphs, but not arbitrarily dense.Comment: To appear in Proc. SoCG 202
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