Coincident-point rigidity in normed planes

TL;DR

This paper investigates the rigidity of bar-joint frameworks in non-Euclidean normed planes with coincident points, establishing a matroid-based characterization and implications for global rigidity.

Contribution

It introduces a matroid characterization of rigidity with coincident points in normed planes and links it to framework independence and global rigidity.

Findings

Matroid independence is equivalent to framework rigidity with coincident points.

A delete-contract characterization of rigidity in normed planes.

Generalized vertex splitting preserves global rigidity in normed planes.

Abstract

A bar-joint framework $(G,p)$ is the combination of a graph $G$ and a map $p$ assigning positions, in some space, to the vertices of $G$. The framework is rigid if every edge-length-preserving continuous motion of the vertices arises from an isometry of the space. We will analyse rigidity when the space is a (non-Euclidean) normed plane and two designated vertices are mapped to the same position. This non-genericity assumption leads us to a count matroid first introduced by Jackson, Kaszanitsky and the third author. We show that independence in this matroid is equivalent to independence as a suitably regular bar-joint framework in a normed plane with two coincident points; this characterises when a regular normed plane coincident-point framework is rigid and allows us to deduce a delete-contract characterisation. We then apply this result to show that an important construction operation (generalised vertex splitting) preserves the stronger property of global rigidity in normed planes and use this to construct rich families of globally rigid graphs when the normed plane is analytic.