When all non-edge distances of a graph realized in $\mathbb{ R}^{ d}$ as a { \em bar-and-joint framework} are generically { \em implied} by the bar (edge) lengths, the graph is said to be { \em rigid} in $\mathbb{ R}^{ d}$. For $d=3$, characterizing rigid graphs, determining implied non-edges and { \em 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., { \em nucleations}, and how to deal with them. In this paper, we give general inductive construction schemes and proof techniques to generate { \em 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 { \em 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 \cite{ sitharam:zhou:tractableADG:2004}, 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 $2D$.
Categories: cs.CG math.CO math.MG