Uniquely Realisable Graphs in Analytic Normed Planes

A framework $(G,p)$ in Euclidean space Eᵈ is globally rigid if it is the unique realisation, up to rigid congruences, of G with the edge lengths of $(G,p)$. Building on key results of Hendrickson [28] and Connelly [14], Jackson and Jordán [29] gave a complete combinatorial characterisation of when a generic framework is global rigidity in E². We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on R² \0\. Specifically, we show that a graph $G=(V,E)$ has an open set of globally rigid realisations in a non-Euclidean analytic normed plane if and only if G is 2-connected and $G-e$ contains 2 edge-disjoint spanning trees for all e∈ E. We also prove that the analogous necessary conditions hold in d-dimensional normed spaces.