Rigid Two-Dimensional Frameworks with Two Coincident Points

Fekete, Zoltán; Jordán, Tibor; Kaszanitzky, Viktória E.

doi:10.1007/s00373-013-1390-0

Cited by 10 publications

(15 citation statements)

References 9 publications

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“…We conclude the introduction with a brief comparison with the more familiar Euclidean case to give context for the reader. Both our characterisation of independence in the coincident-point normed plane rigidity matroid and our delete-contract characterisation are precise analogues of results obtained by Fekete, Jordán and Kaszanitzky for the Euclidean case [11]. Furthermore, in the Euclidean case generic global rigidity in the plane is completely characterised [15].…”

Section: Introductionsupporting

confidence: 73%

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Coincident-point rigidity in normed planes

2023

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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 non-Euclidean normed plane with two coincident points; this characterises when a regular non-Euclidean 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 non-Euclidean normed planes and use this to construct rich families of globally rigid graphs when the non-Euclidean normed plane is analytic.

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Section: Introductionsupporting

confidence: 73%

“…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 [11,13] and applied to global rigidity there [4], as well as for frameworks on surfaces [16].…”

Section: Introductionmentioning

confidence: 99%

Coincident-point rigidity in normed planes

2023

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“…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%

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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“…Another example are the bipartite graphs: Dixon [2] has shown that there are two types of flexible realizations (see also [7,9,11]). Flexible instances of graphs in different context are as well considered in [3,5,9].…”

Section: Introductionmentioning

confidence: 99%

Graphs with Flexible Labelings

Grasegger

Legerský

Schicho

2018

Discrete Comput Geom

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For a flexible labeling of a graph, it is possible to construct infinitely many non-equivalent realizations keeping the distances of connected points constant. We give a combinatorial characterization of graphs that have flexible labelings, possibly non-generic. The characterization is based on colorings of the edges with restrictions on the cycles. Furthermore, we give necessary criteria and sufficient ones for the existence of such colorings.

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Rigid Two-Dimensional Frameworks with Two Coincident Points

Cited by 10 publications

References 9 publications

Coincident-point rigidity in normed planes

Coincident-point rigidity in normed planes

Infinitesimal Rigidity in Normed Planes

Graphs with Flexible Labelings

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