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We describe a very simple condition that is necessary for the universal rigidity of a complete bipartite framework $$(K(n,m),\\mathbf{p},\\mathbf{q})$$(K(n,m),p,q). This condition is also sufficient for universal rigidity under a variety of weak assumptions, such as general position. Even without any of these assumptions, in complete generality, we extend these ideas to obtain an efficient algorithm, based on a sequence of linear programs, that determines whether an input framework of a complete bipartite graph is universally rigid or not.

Universal Rigidity of Complete Bipartite Graphs