Motions of grid-like reflection frameworks

TL;DR

This paper provides combinatorial characterizations of symmetric and anti-symmetric infinitesimal rigidity in 2D reflection frameworks under specific norms, using graph decompositions and sparsity counts.

Contribution

It introduces new combinatorial criteria for rigidity of reflection-symmetric frameworks in quadrilateral normed planes, linking geometric properties to graph theory.

Findings

Characterizations in terms of monochrome subgraph decompositions

Sparsity counts for signed quotient graphs

Recursive construction sequences for frameworks

Abstract

Combinatorial characterisations are obtained of symmetric and anti-symmetric infinitesimal rigidity for two-dimensional frameworks with reflectional symmetry in the case of norms where the unit ball is a quadrilateral and where the reflection acts freely on the vertex set. At the framework level, these characterisations are given in terms of induced monochrome subgraph decompositions, and at the graph level they are given in terms of sparsity counts and recursive construction sequences for the corresponding signed quotient graphs.