Universal rigidity on the line, point order

We show that universal rigidity of a generic bar-joint framework (G,p) in R1 depends on more than the ordering of the vertices. In particular, we construct examples of 1-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 Jordán and Nguyen. Underlying our examples are insights about how universal rigidity behaves under projections. Using these ideas, we also give a simple proof that universal rigidity is invariant under affine transformations.

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