Symmetric isostatic frameworks with $\ell^1$ or $\ell^\infty$ distance constraints

TL;DR

This paper provides combinatorial characterisations for minimal rigidity of symmetric 2D bar-joint frameworks under 1 or 9 distance constraints, using symmetric tree packings and inductive constructions.

Contribution

It introduces new combinatorial criteria and inductive construction schemes for symmetric frameworks with 1 or 9 constraints, extending rigidity theory.

Findings

Characterisations in terms of symmetric tree packings

Use of Henneberg-type inductive schemes

Applicable to 1 and 9 distance constraints

Abstract

Combinatorial characterisations of minimal rigidity are obtained for symmetric 2-dimensional bar-joint frameworks with either $\ell^1$ or $\ell^\infty$ distance constraints. The characterisations are expressed in terms of symmetric tree packings and the number of edges fixed by the symmetry operations. The proof uses new Henneberg-type inductive construction schemes.