On universally rigid frameworks on the line

Abstract

A $d$-dimensional bar-and-joint framework $(G,p)$ with underlying graph $G$ is called universally rigid if all realizations of $G$ with the same edge lengths, in all dimensions, are congruent to $(G,p)$. We give a complete characterization of universally rigid one-dimensional bar-and-joint frameworks in general position with a complete bipartite underlying graph. We show that the only bipartite graph for which all generic $d$-dimensional realizations are universally rigid is the complete graph on two vertices, for all $d\geq 1$. We also discuss several open questions concerning generically universally rigid graphs and the universal rigidity of general frameworks on the line.