Which graphs are rigid in [Formula: see text] ?

We present three results which support the conjecture that a graph is minimally rigid in d-dimensional [Formula: see text] -space, where [Formula: see text] and [Formula: see text] , if and only if it is (d, d)-tight. Firstly, we introduce a graph bracing operation which preserves independence in the generic rigidity matroid when passing from [Formula: see text] to [Formula: see text] . We then prove that every (d, d)-sparse graph with minimum degree at most [Formula: see text] and maximum degree at most [Formula: see text] is independent in [Formula: see text] . Finally, we prove that every triangulation of the projective plane is minimally rigid in [Formula: see text] . A catalogue of rigidity preserving graph moves is also provided for the more general class of strictly convex and smooth normed spaces and we show that every triangulation of the sphere is independent for 3-dimensional spaces in this class.