Uniquely realisable graphs in analytic normed planes

TL;DR

This paper characterizes when graphs are globally rigid in non-Euclidean analytic normed planes, extending known Euclidean results to a broader class of geometric spaces.

Contribution

It provides a complete combinatorial characterization of global rigidity in analytic normed planes, generalizing prior Euclidean results.

Findings

Graph G is globally rigid iff it is 2-connected and G-e contains 2 edge-disjoint spanning trees for all edges e.

Main technical tool is a recursive construction of 2-connected, redundantly rigid graphs in these spaces.

Necessary conditions for global rigidity are established in higher-dimensional analytic normed spaces.

Abstract

A bar-joint framework $(G,p)$ in the Euclidean space $\mathbb{E}^d$ is globally rigid if it is the unique realisation, up to rigid congruences, of $G$ in $\mathbb{E}^d$ with the edge lengths of $(G,p)$. Building on key results of Hendrickson and Connelly, Jackson and Jord\'{a}n gave a complete combinatorial characterisation of when a generic framework is global rigidity in $\mathbb{E}^2$. We prove an analogous result when the Euclidean norm is replaced by any norm that is analytic on $\mathbb{R}^2 \setminus \{0\}$. More precisely, we show that a graph $G=(V,E)$ is globally rigid 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 $e\in E$. The main technical tool is a recursive construction of 2-connected and redundantly rigid graphs in analytic normed planes. We also obtain some sufficient conditions for global rigidity as corollaries of our main result and prove that the analogous necessary conditions hold in $d$-dimensional analytic normed spaces.