Symmetric Isostatic Frameworks with $\ell^1$ or $\ell^\infty$ 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.