Rigid Cylindrical Frameworks with Two Coincident Points

Jackson, Bill; Kaszanitzky, Viktória E.; Nixon, Anthony

doi:10.1007/s00373-018-1983-8

Cited by 8 publications

(19 citation statements)

References 12 publications

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“…With the geometric results of the previous section in hand, we can use the combinatorics of [14] to prove the difficult sufficiency direction of our main result on coincident frameworks. We begin with the following result which can be extracted from the proof of [14,Theorem 4].…”

Section: Characterising Coincident Point Independencementioning

confidence: 99%

“…The difficulty that already arises in this context shows how necessary the genericity assumption in those papers really was. Frameworks with coincident points have been considered in the Euclidean context [10,12] and applied to global rigidity there [4], as well as for frameworks on surfaces [14].…”

Section: Introductionmentioning

confidence: 99%

“…Here we provide a detailed geometric analysis of the effect of certain graph operations on the rigidity of a coincident point framework in a normed plane. In Section 4 we combine these geometric results with combinatorial results of [14] to establish a purely combinatorial characterisation of independence in the 'coincident point normed plane rigidity matroid' and we deduce from this a delete-contract characterisation of coincident point rigidity in any strictly convex normed plane. In Section 5 we provide our other main results; these concern global rigidity.…”

Section: Introductionmentioning

confidence: 99%

“…It was shown in [14] that the edge sets of the uv-sparse subgraphs of G form the independent sets of a matroid, and when |V | ≥ 5 this matroid has rank 2|V | − 2.…”

Section: Introductionmentioning

confidence: 99%

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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: Characterising Coincident Point Independencementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…It was shown in [14] that the edge sets of the uv-sparse subgraphs of G form the independent sets of a matroid, and when |V | ≥ 5 this matroid has rank 2|V | − 2.…”

Section: Introductionmentioning

confidence: 99%

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Infinitesimal Rigidity in Normed Planes

Dewar¹

2020

SIAM J. Discrete Math.

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“…We next consider the generalised vertex splitting operation. In [12,Theorem 5.2] it was proved that the standard vertex splitting operation, with the additional assumption that the new graph is rigid when we delete the bridging edge (i.e. the edge joining the two copies of the split vertex), preserves generic global rigidity on the cylinder.…”

Section: Maximum Rank Equilibrium Stressesmentioning

confidence: 99%

Global rigidity of generic frameworks on the cylinder

Jackson

Nixon

2019

Journal of Combinatorial Theory, Series B

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We show that a generic framework (G, p) on the cylinder is globally rigid if and only if G is a complete graph on at most four vertices or G is both redundantly rigid and 2-connected. To prove the theorem we also derive a new recursive construction of circuits in the simple (2, 2)-sparse matroid, and a characterisation of rigidity for generic frameworks on the cylinder when a single designated vertex is allowed to move off the cylinder.Date: October 16, 2018. 2010 Mathematics Subject Classification. 52C25, 05C10 and 53A05.

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Vertex Splitting, Coincident Realisations, and Global Rigidity of Braced Triangulations

Jackson

Cruickshank²

2022

Discrete Comput Geom

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Rigid Cylindrical Frameworks with Two Coincident Points

Cited by 8 publications

References 12 publications

Infinitesimal Rigidity in Normed Planes

Infinitesimal Rigidity in Normed Planes

Global rigidity of generic frameworks on the cylinder

Vertex Splitting, Coincident Realisations, and Global Rigidity of Braced Triangulations

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