2,505 research outputs found
On the Number of Embeddings of Minimally Rigid Graphs
Rigid frameworks in some Euclidian space are embedded graphs having a unique
local realization (up to Euclidian motions) for the given edge lengths,
although globally they may have several. We study the number of distinct planar
embeddings of minimally rigid graphs with vertices. We show that, modulo
planar rigid motions, this number is at most . We also exhibit several families which realize lower bounds of the order
of , and .
For the upper bound we use techniques from complex algebraic geometry, based
on the (projective) Cayley-Menger variety over the complex numbers . In this context, point configurations
are represented by coordinates given by squared distances between all pairs of
points. Sectioning the variety with hyperplanes yields at most
zero-dimensional components, and one finds this degree to be
. The lower bounds are related to inductive
constructions of minimally rigid graphs via Henneberg sequences.
The same approach works in higher dimensions. In particular we show that it
leads to an upper bound of for the number of spatial embeddings
with generic edge lengths of the 1-skeleton of a simplicial polyhedron, up to
rigid motions
Characterization and enumeration of toroidal K_{3,3}-subdivision-free graphs
We describe the structure of 2-connected non-planar toroidal graphs with no
K_{3,3}-subdivisions, using an appropriate substitution of planar networks into
the edges of certain graphs called toroidal cores. The structural result is
based on a refinement of the algorithmic results for graphs containing a fixed
K_5-subdivision in [A. Gagarin and W. Kocay, "Embedding graphs containing
K_5-subdivisions'', Ars Combin. 64 (2002), 33-49]. It allows to recognize these
graphs in linear-time and makes possible to enumerate labelled 2-connected
toroidal graphs containing no K_{3,3}-subdivisions and having minimum vertex
degree two or three by using an approach similar to [A. Gagarin, G. Labelle,
and P. Leroux, "Counting labelled projective-planar graphs without a
K_{3,3}-subdivision", submitted, arXiv:math.CO/0406140, (2004)].Comment: 18 pages, 7 figures and 4 table
The obstructions for toroidal graphs with no 's
Forbidden minors and subdivisions for toroidal graphs are numerous. We
consider the toroidal graphs with no -subdivisions that coincide with
the toroidal graphs with no -minors. These graphs admit a unique
decomposition into planar components and have short lists of obstructions. We
provide the complete lists of four forbidden minors and eleven forbidden
subdivisions for the toroidal graphs with no 's and prove that the
lists are sufficient.Comment: 10 pages, 7 figures, revised version with additional detail
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