556 research outputs found
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
Shortest Reconfiguration of Perfect Matchings via Alternating Cycles
Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar
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
Unital associahedra
We construct a topological cellular operad such that the algebras over its
cellular chains are the homotopy unital A-infinity algebras of
Fukaya-Oh-Ohta-Ono.Comment: 22 pages, EPS color figure
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