Uniquely realisable graphs in polyhedral normed spaces

TL;DR

This work analyzes global rigidity of frameworks in polyhedral normed spaces, focusing on how edge-length constraints interact with non-Euclidean distance. It introduces directed colourings and the rigidity matrices $M(G,\phi)$ to obtain exact characterisations and algorithms for global rigidity, proving NP-hardness in polyhedral spaces while delivering efficient tests for well-positioned cases and the $\ell_\infty^2$ plane. A central contribution is the $\ell_\infty^d$-specific theory, including a complete 2D characterisation: a generic framework in $\ell_\infty^2$ is globally rigid iff the graph is $\mathcal{M}(2,2)$-connected, and Hendrickson’s condition can fail in this setting. The paper also shows that global rigidity in $\ell_\infty^d$ can be inferred from rigidity in sequences of $\ell_p^d$ spaces and from strong directed colourings, yielding practical sufficient conditions and highlighting structural gaps between global rigidity and redundant rigidity in non-Euclidean norms.

Abstract

A framework (a straight-line embedding of a graph into a normed space allowing edges to cross) is globally rigid if any other framework with the same edge lengths with respect to the chosen norm is an isometric copy. We investigate global rigidity in polyhedral normed spaces: normed spaces where the unit ball is a polytope. We first provide a deterministic algorithm for checking whether or not a framework in a polyhedral normed space is globally rigid. After showing that determining if a framework is globally rigid is NP-Hard, we then provide necessary conditions for global rigidity for generic frameworks. We obtain stronger results for generic frameworks in $\ell_\infty^d$ (the vector space $\mathbb{R}^d$ equipped with the $\ell_\infty$ metric) including an exact characterisation of global rigidity when $d=2$, and an easily-computable sufficient condition for global rigidity using edge colourings. Our 2-dimensional characterisation also has a surprising consequence: Hendrickson's global rigidity condition fails for generic frameworks in $\ell_\infty^2$.

Figures (4)

• Figure 1: A flexible framework in $\ell_\infty^2$ which is rigid under the standard Euclidean metric. This is solely due to the lack of isometry of $\ell_\infty^2$ which fixes the centre vertex and takes the upper vertex to any point on the dashed square.
• Figure 2: The double-banana graph. The graph is $(3,6)$-tight, but clearly has no infinitesimally rigid realisations in 3-dimensional Euclidean space due to the vertex cut pair acting like a hinge.
• Figure 3: Construction described in \ref{['thm:nphard']} when $d=2$ and $G$ is the graph with vertices $v_0,v_1,v_2$ and edges $v_0v_1,v_1v_2$. The original framework $(G,p)$ is given in black (vertices from left to right: $v_0,v_1,v_2$), and the subframework of $(G',p')$ induced on $S$ is given in grey. Edges between vertices in $V$ and $S$ are given in blue.
• Figure 4: The monochromatic subgraphs of the framework $(G,p)$ given in \ref{['ex:oct']}: blue represents an edge with corresponding face $(\pm1,0)$, red represents an edge with corresponding face $(0,\pm1)$.

Theorems & Definitions (92)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 2.1: Dewar Dewar22
• Corollary 2.2: Dewar21
• Theorem 2.3: Pollaczek-Geiringer PollaczekGeiringer