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When all non-edge distances of a graph realized in Rd as a bar-and-joint framework are generically implied by the bar (edge) lengths, the graph is said to be rigid in Rd. For d = 3, characterizing rigid graphs, determining implied non-edges and dependent edge sets remains an elusive, long-standing open problem.

One obstacle is to determine when implied non-edges can exist without non-trivial rigid induced subgraphs, i.e., nucleations, and how to deal with them.

In this paper, we give general inductive construction schemes and proof techniques to generate nucleation-free graphs (i.e., graphs without any nucleation) with implied non-edges. As a consequence, we obtain (a) dependent graphs in 3D that have no nucleation; and (b) 3D nucleation-free rigidity circuits, i.e., minimally dependent edge sets in d = 3. It additionally follows that true rigidity is strictly stronger than a tractable approximation to rigidity given by Sitharam and Zhou [16], based on an inductive combinatorial characterization.

As an independently interesting byproduct, we obtain a new inductive construction for independent graphs in 3D. Currently, very few such inductive constructions are known, in contrast to 2D.


Another paper by the same title was presented by the authors at the 21st Canadian Conference on Computational Geometry (CCCG2009). The full text of that paper is available at