Which graphs are rigid in lpd?

Abstract

We present three results which support the conjecture that a graph is minimally rigid in d-dimensional ℓp-space, where p∈(1,∞) and 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 ℓdp to ℓd+1p. We then prove that every (d, d)-sparse graph with minimum degree at most d+1 and maximum degree at most d+2 is independent in ℓdp. Finally, we prove that every triangulation of the projective plane is minimally rigid in ℓ3p. 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.