Motions of grid-like reflection frameworks

D Kitson,B Schulze

doi:10.1016/j.jsc.2018.02.004

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.

Highlights

Recent work in geometric rigidity has seen an analysis of frameworks in which the standard Euclidean norm is replaced by a non-Euclidean norm
We consider two-dimensional frameworks and norms for which the unit ball is a quadrilateral. Such frameworks are grid-like in the sense that the allowable motions constrain vertices adjacent to any pinned vertex to move along the boundary of a quadrilateral which is centred at the pinned vertex and obtained from the unit ball by translation and dilation
The problem of maintaining rigid formations of autonomous agents is a well-known application of geometric rigidity theory and its associated “pebble game” algorithms

Summary

Introduction

Recent work in geometric rigidity has seen an analysis of frameworks in which the standard Euclidean norm is replaced by a non-Euclidean norm (see [6, 7, 8, 9]). The third aim, which is in the spirit of Laman’s theorem (see [10, 20, 22]), is to provide complete characterisations for graphs which admit placements as rigid grid-like frameworks with reflectional symmetry These characterisations provide the sufficiency direction for the necessary counting conditions derived in the general theory of Section 2. We note that these matroidal counts can be checked in polynomial time using a straightforward adaptation of the algorithm described in [4, Sect. 10] (see [1])

Z2-symmetric frameworks in normed spaces

Grid-like frameworks with reflectional symmetry

Further remarks