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

Sean Dewar (), Derek Kitson () and Anthony Nixon ()
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Sean Dewar: Austrian Academy of Sciences
Derek Kitson: Lancaster University
Anthony Nixon: Lancaster University

Journal of Global Optimization, 2022, vol. 83, issue 1, No 4, 49-71

Abstract: 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 ≠ 2 , if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from $$\ell _p^d$$ ℓ p d to $$\ell _p^{d+1}$$ ℓ p d + 1 . We then prove that every (d, d)-sparse graph with minimum degree at most $$d+1$$ d + 1 and maximum degree at most $$d+2$$ d + 2 is independent in $$\ell _p^d$$ ℓ p d . Finally, we prove that every triangulation of the projective plane is minimally rigid in $$\ell _p^3$$ ℓ p 3 . A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.

Keywords: Bar-joint framework; Infinitesimal rigidity; Rigidity matroid; Normed spaces; 52C25; 05C50 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-021-01008-z

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