Which graphs are rigid in $$\ell _p^d$$?

Dewar, Sean; Kitson, Derek; Nixon, Anthony

doi:10.1007/s10898-021-01008-z

Cited by 7 publications

(11 citation statements)

References 13 publications

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“…In the last decade, work on rigidity has been generalised to various non-Euclidean normed spaces (e.g. [5,8,9,16,17]). All of these results concern characterising the combinatorial nature of the 'generic' behaviour of frameworks.…”

Section: Introductionmentioning

confidence: 99%

Infinitesimal Rigidity in Normed Planes

Dewar¹

2020

SIAM J. Discrete Math.

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A bar-joint framework (G, p) is the combination of a graph G and a map p assigning positions, in some space, to the vertices of G. The framework is rigid if every edge-lengthpreserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.

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Section: Introductionmentioning

confidence: 99%

Infinitesimal Rigidity in Normed Planes

Dewar¹

2020

SIAM J. Discrete Math.

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“…Triangulated spheres with a single brace have previously been studied in [20] in relation to redundant rigidity and more recently in [7,11] in relation to global rigidity. The case b = 2 is more involved and in Section 4 we show that there are exactly five distinct irreducibles (see Figures 3,4,5,6,7).…”

Section: Introductionmentioning

confidence: 93%

“…Of course, topological vertex splitting is a special case of (3-dimensional) vertex splitting for graphs, and it is known that vertex splitting for graphs preserves rigidity for generic frameworks in a wide variety of settings [21,4,7,8]. Therefore it is natural to look for geometric rigidity applications of Theorem 20.…”

Section: Theorem 14 Any Irreducible Braced Triangulation With Two Bra...mentioning

confidence: 99%

“…Let T (R 4 ) denote the real linear space of infinitesimal rigid motions of the Euclidean space R 4 . Recall that each infinitesimal rigid motion η ∈ T (R 4 ) is an affine map η : R 4 → R 4 of the form η(x) = B(x) + c where the linear part B is a 4 × 4 skew-symmetric matrix and the translational part c is a vector in R 4 .…”

Section: The Hypercylinder Inmentioning

confidence: 99%

“…Two major new research strands in geometric rigidity are the rigidity analyses of barjoint frameworks in Euclidean 3-space whose joints are constrained to move on a surface (such as a cylinder or surface of revolution) [8,9,16,17] and of bar-joint frameworks in non-Euclidean normed spaces [4,12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

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Braced triangulations and rigidity

Cruickshank¹,

Kastis²,

Kitson³

et al. 2021

Preprint

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We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b = 1 or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R 4 and a class of mixed norms on R 3 . 2020 Mathematics Subject Classification. 52C25, 05C10, 46B20. E.K. and D.K.

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The Rigidity of Infinite Graphs II

Kitson¹,

Power

2022

Graphs and Combinatorics

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Inductive constructions are established for countably infinite simple graphs which have minimally rigid locally generic placements in R 2 . This generalises a well-known result of Henneberg for generically rigid finite graphs. Inductive methods are also employed in the determination of the infinitesimal flexibility dimension of countably infinite graphs associated with infinitely faceted convex polytopes in R 3 . In particular, a generalisation of Cauchy's rigidity theorem is obtained.

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Which graphs are rigid in $$\ell _p^d$$?

Abstract: We present three results which support the conjecture that a graph is minimally rigid in d-dimensional $$\ell _p$$ ℓ p -space, where $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) and $$p\not =2$$ p ≠ … Show more

Cited by 7 publications

References 13 publications

Infinitesimal Rigidity in Normed Planes

Infinitesimal Rigidity in Normed Planes

Braced triangulations and rigidity

The Rigidity of Infinite Graphs II

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