Global rigidity of generic frameworks on the cylinder

TL;DR

This paper characterizes when generic frameworks on a cylinder are globally rigid, establishing conditions based on graph properties and introducing new recursive constructions and characterizations for such frameworks.

Contribution

It provides a complete characterization of global rigidity for frameworks on the cylinder and introduces new recursive methods for circuits in the $(2,2)$-sparse matroid.

Findings

A framework on the cylinder is globally rigid if and only if it is either a complete graph on at most four vertices or both redundantly rigid and 2-connected.

Derived a new recursive construction of circuits in the simple $(2,2)$-sparse matroid.

Characterized rigidity for frameworks on the cylinder with a movable vertex off the cylinder.

Abstract

We show that a generic framework $(G,p)$ on the cylinder is globally rigid if and only if $G$ is a complete graph on at most four vertices or $G$ is both redundantly rigid and $2$-connected. To prove the theorem we also derive a new recursive construction of circuits in the simple $(2,2)$-sparse matroid, and a characterisation of rigidity for generic frameworks on the cylinder when a single designated vertex is allowed to move off the cylinder.