Universal rigidity on the line, point order

TL;DR

This paper demonstrates that in one-dimensional frameworks, universal rigidity depends on factors beyond vertex order, providing counterexamples that challenge previous assumptions.

Contribution

It shows that vertex order alone does not determine universal rigidity in the line, answering a question posed by Jordan and Nguyen.

Findings

Counterexamples of frameworks with same graph and order but different rigidity

Universal rigidity depends on more than just vertex order in 1D frameworks

Challenges previous assumptions about rigidity criteria

Abstract

We show that universal rigidity of a generic bar and joint framework (G,p) in the line depends on more than the ordering of the vertices. In particular, we construct examples of one-dimensional generic frameworks with the same graph and ordering of the vertices, such that one is universally rigid and one is not. This answers, in the negative, a question of Jordan and Nguyen.