Coincident-point rigidity in normed planes

Dewar, Sean and Hewetson, John and Nixon, Anthony (2023) Coincident-point rigidity in normed planes. Ars Mathematica Contemporanea, 24 (1). ISSN 1855-3966

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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-length-preserving 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.

Item Type:
Journal Article
Journal or Publication Title:
Ars Mathematica Contemporanea
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Research Output Funding/yes_externally_funded
?? bar-joint frameworkglobal rigiditynon-euclidean frameworkcount matroidrecursive constructionnormed spacesanalytic normyes - externally fundedtheoretical computer sciencealgebra and number theorydiscrete mathematics and combinatoricsgeometry and topology ??
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Deposited On:
25 Apr 2023 10:40
Last Modified:
15 Jul 2024 23:41