Section:
New Results
Mixed volume and distance geometry techniques for
counting Euclidean embeddings of rigid graphs.
A graph is called generically minimally rigid
in if, for any choice of sufficiently
generic edge lengths, it can be embedded in
in a finite number of distinct ways, modulo rigid
transformations.
In [37] we deal with the problem of
determining tight bounds on the number of such
embeddings, as a function of the number of
vertices. The study of rigid graphs is motivated by
numerous applications, mostly in robotics,
bioinformatics, and architecture. We capture
embeddability by polynomial systems with suitable
structure, so that their mixed volume, which bounds
the number of common roots, yields interesting upper
bounds on the number of embeddings. We explore
different polynomial formulations so as to reduce
the corresponding mixed volume, namely by
introducing new variables that remove certain
spurious roots, and by applying the theory of
distance geometry. We focus on and ,
where Laman graphs and 1-skeleta of convex
simplicial polyhedra, respectively, admit inductive
Henneberg constructions. Our implementation yields
upper bounds for in and ,
which reduce the existing gaps and lead to tight
bounds for in both and ; in
particular, we describe the recent settlement of the
case of Laman graphs with 7 vertices. We also
establish the first lower bound in of about
, where denotes the number of
vertices.