227 research outputs found
Natural realizations of sparsity matroids
A hypergraph G with n vertices and m hyperedges with d endpoints each is
(k,l)-sparse if for all sub-hypergraphs G' on n' vertices and m' edges, m'\le
kn'-l. For integers k and l satisfying 0\le l\le dk-1, this is known to be a
linearly representable matroidal family.
Motivated by problems in rigidity theory, we give a new linear representation
theorem for the (k,l)-sparse hypergraphs that is natural; i.e., the
representing matrix captures the vertex-edge incidence structure of the
underlying hypergraph G.Comment: Corrected some typos from the previous version; to appear in Ars
Mathematica Contemporane
Rigidity of frameworks on expanding spheres
A rigidity theory is developed for bar-joint frameworks in
whose vertices are constrained to lie on concentric -spheres with
independently variable radii. In particular, combinatorial characterisations
are established for the rigidity of generic frameworks for with an
arbitrary number of independently variable radii, and for with at most
two variable radii. This includes a characterisation of the rigidity or
flexibility of uniformly expanding spherical frameworks in .
Due to the equivalence of the generic rigidity between Euclidean space and
spherical space, these results interpolate between rigidity in 1D and 2D and to
some extent between rigidity in 2D and 3D. Symmetry-adapted counts for the
detection of symmetry-induced continuous flexibility in frameworks on spheres
with variable radii are also provided.Comment: 22 pages, 2 figures, updated reference
One brick at a time: a survey of inductive constructions in rigidity theory
We present a survey of results concerning the use of inductive constructions
to study the rigidity of frameworks. By inductive constructions we mean simple
graph moves which can be shown to preserve the rigidity of the corresponding
framework. We describe a number of cases in which characterisations of rigidity
were proved by inductive constructions. That is, by identifying recursive
operations that preserved rigidity and proving that these operations were
sufficient to generate all such frameworks. We also outline the use of
inductive constructions in some recent areas of particularly active interest,
namely symmetric and periodic frameworks, frameworks on surfaces, and body-bar
frameworks. We summarize the key outstanding open problems related to
inductions.Comment: 24 pages, 12 figures, final versio
A necessary condition for generic rigidity of bar-and-joint frameworks in -space
A graph is -sparse if each subset with induces at most edges in . Maxwell showed in
1864 that a necessary condition for a generic bar-and-joint framework with at
least vertices to be rigid in is that should have a
-sparse subgraph with edges. This necessary
condition is also sufficient when but not when . Cheng and
Sitharam strengthened Maxwell's condition by showing that every maximal
-sparse subgraph of should have edges when
. We extend their result to all .Comment: There was an error in the proof of Theorem 3.3(b) in version 1 of
this paper. A weaker statement was proved in version 2 and then used to
derive the main result Theorem 4.1 when . The proof technique was
subsequently refined in collaboration with Hakan Guler to extend this result
to all in Theorem 3.3 of version
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