4 research outputs found
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
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)
On the Maximal Number of Real Embeddings of Spatial Minimally Rigid Graphs
The number of embeddings of minimally rigid graphs in RD 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 R-3. We modify a commonly used parametric semi-algebraic
formulation that exploits the CayleyMenger determinant to minimize the 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 R-3, 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 R-3