Global rigidity of generic frameworks on the cylinder

Jackson, Bill; Nixon, Anthony

doi:10.1016/j.jctb.2019.03.002

Cited by 16 publications

(14 citation statements)

References 22 publications

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“…Moreover any graph obtained from either of these by a sequence of degree 3 vertex additions (i.e., add a vertex and join it to three other vertices) and edge additions is globally rigid. We next increase this class of graphs with the following construction operation introduced in [15]. A generalised vertex split, is defined as follows.…”

Section: Proposition 52 ([9]mentioning

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: Proposition 52 ([9]mentioning

confidence: 99%

Infinitesimal Rigidity in Normed Planes

Dewar¹

2020

SIAM J. Discrete Math.

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“…We have already described the case of the sphere in detail from a different viewpoint. For other surfaces, such as the cylinder, see [45,83] for rigidity and [84] for global rigidity.…”

Section: Linear Constraints As Slidersmentioning

confidence: 99%

Rigidity through a Projective Lens

2021

Self Cite

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In this paper, we offer an overview of a number of results on the static rigidity and infinitesimal rigidity of discrete structures which are embedded in projective geometric reasoning, representations, and transformations. Part I considers the fundamental case of a bar–joint framework in projective d-space and places particular emphasis on the projective invariance of infinitesimal rigidity, coning between dimensions, transfer to the spherical metric, slide joints and pure conditions for singular configurations. Part II extends the results, tools and concepts from Part I to additional types of rigid structures including body-bar, body–hinge and rod-bar frameworks, all drawing on projective representations, transformations and insights. Part III widens the lens to include the closely related cofactor matroids arising from multivariate splines, which also exhibit the projective invariance. These are another fundamental example of abstract rigidity matroids with deep analogies to rigidity. We conclude in Part IV with commentary on some nearby areas.

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“…In several applications in rigidity theory (see [11,23]), it must be assumed that all considered graphs are simple, that is, have no loops nor parallel edges. Hence only those redundant augmentations are appropriate that maintain this property, that is, the input as well as the output graph is simple.…”

Section: Simple Graphsmentioning

confidence: 99%

Sparse graphs and an augmentation problem

Király

Mihálykó

2021

Math. Program.

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For two integers $$k>0$$ k > 0 and $$\ell $$ ℓ , a graph $$G=(V,E)$$ G = ( V , E ) is called $$(k,\ell )$$ ( k , ℓ ) -tight if $$|E|=k|V|-\ell $$ | E | = k | V | - ℓ and $$i_G(X)\le k|X|-\ell $$ i G ( X ) ≤ k | X | - ℓ for each $$X\subseteq V$$ X ⊆ V for which $$i_G(X)\ge 1$$ i G ( X ) ≥ 1 , where $$i_G(X)$$ i G ( X ) denotes the number of induced edges by X. G is called $$(k,\ell )$$ ( k , ℓ ) -redundant if $$G-e$$ G - e has a spanning $$(k,\ell )$$ ( k , ℓ ) -tight subgraph for all $$e\in E$$ e ∈ E . We consider the following augmentation problem. Given a graph $$G=(V,E)$$ G = ( V , E ) that has a $$(k,\ell )$$ ( k , ℓ ) -tight spanning subgraph, find a graph $$H=(V,F)$$ H = ( V , F ) with the minimum number of edges, such that $$G\cup H$$ G ∪ H is $$(k,\ell )$$ ( k , ℓ ) -redundant. We give a polynomial algorithm and a min-max theorem for this augmentation problem when the input is $$(k,\ell )$$ ( k , ℓ ) -tight. For general inputs, we give a polynomial algorithm when $$k\ge \ell $$ k ≥ ℓ and show the NP-hardness of the problem when $$k<\ell $$ k < ℓ . Since $$(k,\ell )$$ ( k , ℓ ) -tight graphs play an important role in rigidity theory, these algorithms can be used to make several types of rigid frameworks redundantly rigid by adding a smallest set of new bars.

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Global rigidity of generic frameworks on the cylinder

Cited by 16 publications

References 22 publications

Infinitesimal Rigidity in Normed Planes

Infinitesimal Rigidity in Normed Planes

Rigidity through a Projective Lens

Sparse graphs and an augmentation problem

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