23 research outputs found
Mixed Volume Techniques for Embeddings of Laman Graphs
Determining the number of embeddings of Laman graph frameworks is an open
problem which corresponds to understanding the solutions of the resulting
systems of equations. In this paper we investigate the bounds which can be
obtained from the viewpoint of Bernstein's Theorem. The focus of the paper is
to provide the methods to study the mixed volume of suitable systems of
polynomial equations obtained from the edge length constraints. While in most
cases the resulting bounds are weaker than the best known bounds on the number
of embeddings, for some classes of graphs the bounds are tight.Comment: Thorough revision of the first version. (13 pages, 4 figures
Lower bounds on the number of realizations of rigid graphs
Computing the number of realizations of a minimally rigid graph is a
notoriously difficult problem. Towards this goal, for graphs that are minimally
rigid in the plane, we take advantage of a recently published algorithm, which
is the fastest available method, although its complexity is still exponential.
Combining computational results with the theory of constructing new rigid
graphs by gluing, we give a new lower bound on the maximal possible number of
(complex) realizations for graphs with a given number of vertices. We extend
these ideas to rigid graphs in three dimensions and we derive similar lower
bounds, by exploiting data from extensive Gr\"obner basis computations
Radically solvable graphs
A 2-dimensional framework is a straight line realisation of a graph in the
Euclidean plane. It is radically solvable if the set of vertex coordinates is
contained in a radical extension of the field of rationals extended by the
squared edge lengths. We show that the radical solvability of a generic
framework depends only on its underlying graph and characterise which planar
graphs give rise to radically solvable generic frameworks. We conjecture that
our characterisation extends to all graphs
On the maximal number of real embeddings of minimally rigid graphs in , and
Rigidity theory studies the properties of graphs that can have rigid
embeddings in a euclidean space or on a sphere and which in
addition satisfy certain edge length constraints. One of the major open
problems in this field is to determine lower and upper bounds on the number of
realizations with respect to a given number of vertices. This problem is
closely related to the classification of rigid graphs according to their
maximal number of real embeddings.
In this paper, we are interested in finding edge lengths that can maximize
the number of real embeddings of minimally rigid graphs in the plane, space,
and on the sphere. We use algebraic formulations to provide upper bounds. To
find values of the parameters that lead to graphs with a large number of real
realizations, possibly attaining the (algebraic) upper bounds, we use some
standard heuristics and we also develop a new method inspired by coupler
curves. We apply this new method to obtain embeddings in . One of
its main novelties is that it allows us to sample efficiently from a larger
number of parameters by selecting only a subset of them at each iteration.
Our results include a full classification of the 7-vertex graphs according to
their maximal numbers of real embeddings in the cases of the embeddings in
and , while in the case of we achieve this
classification for all 6-vertex graphs. Additionally, by increasing the number
of embeddings of selected graphs, we improve the previously known asymptotic
lower bound on the maximum number of realizations. The methods and the results
concerning the spatial embeddings are part of the proceedings of ISSAC 2018
(Bartzos et al, 2018)
Elimination for generic sparse polynomial systems
We present a new probabilistic symbolic algorithm that, given a variety
defined in an n-dimensional affine space by a generic sparse system with fixed
supports, computes the Zariski closure of its projection to an l-dimensional
coordinate affine space with l < n. The complexity of the algorithm depends
polynomially on combinatorial invariants associated to the supports.Comment: 22 page
On the maximal number of real embeddings of spatial minimally rigid graphs
The number of embeddings of minimally rigid graphs in is (by
definition) finite, modulo rigid transformations, for every generic choice of
edge lengths. Even though various approaches have been proposed to compute it,
the gap between upper and lower bounds is still enormous. Specific values and
its asymptotic behavior are major and fascinating open problems in rigidity
theory. Our work considers the maximal number of real embeddings of minimally
rigid graphs in . We modify a commonly used parametric
semi-algebraic formulation that exploits the Cayley-Menger determinant to
minimize the {\em a priori} number of complex embeddings, where the parameters
correspond to edge lengths. To cope with the huge dimension of the parameter
space and find specializations of the parameters that maximize the number of
real embeddings, we introduce a method based on coupler curves that makes the
sampling feasible for spatial minimally rigid graphs.
Our methodology results in the first full classification of the number of
real embeddings of graphs with 7 vertices in , which was the
smallest open case. Building on this and certain 8-vertex graphs, we improve
the previously known general lower bound on the maximum number of real
embeddings in