7 research outputs found
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
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
New upper bounds for the number of embeddings of minimally rigid graphs
By definition, a rigid graph in (or on a sphere) has a finite
number of embeddings up to rigid motions for a given set of edge length
constraints. These embeddings are related to the real solutions of an algebraic
system. Naturally, the complex solutions of such systems extend the notion of
rigidity to . A major open problem has been to obtain tight upper
bounds on the number of embeddings in , for a given number
of vertices, which obviously also bound their number in .
Moreover, in most known cases, the maximal numbers of embeddings in
and coincide. For decades, only the trivial bound
of was known on the number of embeddings.Recently, matrix
permanent bounds have led to a small improvement for . This work
improves upon the existing upper bounds for the number of embeddings in
and , by exploiting outdegree-constrained orientations on a
graphical construction, where the proof iteratively eliminates vertices or
vertex paths. For the most important cases of and , the new bounds
are and , respectively. In general, the
recent asymptotic bound mentioned above is improved by a factor of . Besides being the first substantial improvement upon a long-standing
upper bound, our method is essentially the first general approach relying on
combinatorial arguments rather than algebraic root counts
On the multihomogeneous Bézout bound on the number of embeddings of minimally rigid graphs
International audienceRigid graph theory is an active area with many open problems, especially regarding embeddings in R^d or other manifolds, and tight upper bounds on their number for a given number of vertices. Our premise is to relate the number of embeddings to that of solutions of a well-constrained algebraic system and exploit progress in the latter domain. In particular, the system's complex solutions naturally extend the notion of real embeddings, thus allowing us to employ bounds on complex roots. We focus on multihomogeneous Bézout (m-Bézout) bounds of algebraic systems since they are fast to compute and rather tight for systems exhibiting structure as in our case. We introduce two methods to relate such bounds to combinatorial properties of minimally rigid graphs in C^d and S^d. The first relates the number of graph orientations to the m-Bézout bound, while the second leverages a matrix permanent formulation. Using these approaches we improve the best known asymptotic upper bounds for planar graphs in dimension 3, and all minimally rigid graphs in dimension d ≥ 5, both in the Euclidean and spherical case. Our computations indicate that m-Bézout bounds are tight for embeddings of planar graphs in S^2 and C36. We exploit Bernstein's second theorem on the exactness of mixed volume, and relate it to the m-Bézout bound by analyzing the associated Newton polytopes. We reduce the number of checks required to verify exactness by an exponential factor, and conjecture further that it suffices to check a linear instead of an exponential number of cases overall