6,553 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
Fat 4-polytopes and fatter 3-spheres
We introduce the fatness parameter of a 4-dimensional polytope P, defined as
\phi(P)=(f_1+f_2)/(f_0+f_3). It arises in an important open problem in
4-dimensional combinatorial geometry: Is the fatness of convex 4-polytopes
bounded?
We describe and analyze a hyperbolic geometry construction that produces
4-polytopes with fatness \phi(P)>5.048, as well as the first infinite family of
2-simple, 2-simplicial 4-polytopes. Moreover, using a construction via finite
covering spaces of surfaces, we show that fatness is not bounded for the more
general class of strongly regular CW decompositions of the 3-sphere.Comment: 12 pages, 12 figures. This version has minor changes proposed by the
second refere
The rigidity of infinite graphs
A rigidity theory is developed for the Euclidean and non-Euclidean placements
of countably infinite simple graphs in R^d with respect to the classical l^p
norms, for d>1 and 1<p<\infty. Generalisations are obtained for the Laman and
Henneberg combinatorial characterisations of generic infinitesimal rigidity for
finite graphs in the Euclidean plane. Also Tay's multi-graph characterisation
of the rigidity of generic finite body-bar frameworks in d-dimensional
Euclidean space is generalised to the non-Euclidean l^p norms and to countably
infinite graphs. For all dimensions and norms it is shown that a generically
rigid countable simple graph is the direct limit of an inclusion tower of
finite graphs for which the inclusions satisfy a relative rigidity property.
For d>2 a countable graph which is rigid for generic placements in R^d may fail
the stronger property of sequential rigidity, while for d=2 the equivalence
with sequential rigidity is obtained from the generalised Laman
characterisations. Applications are given to the flexibility of non-Euclidean
convex polyhedra and to the infinitesimal and continuous rigidity of compact
infinitely-faceted simplicial polytopes.Comment: 51 page
Pseudograph associahedra
Given a simple graph G, the graph associahedron KG is a simple polytope whose
face poset is based on the connected subgraphs of G. This paper defines and
constructs graph associahedra in a general context, for pseudographs with loops
and multiple edges, which are also allowed to be disconnected. We then consider
deformations of pseudograph associahedra as their underlying graphs are altered
by edge contractions and edge deletions.Comment: 25 pages, 22 figure
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