A necessary condition for generic rigidity of bar-and-joint frameworks in d -space

A graph G=(V,E) is d-sparse if each subset X⊆V with |X|≥d induces at most d|X|−(d+12) edges in G. Maxwell showed in 1864 that a necessary condition for a generic bar-and-joint framework with at least d+1 vertices to be rigid in Rd is that G should have a d-sparse subgraph with d|X|−(d+12) edges. This necessary condition is also sufficient when d=1,2 but not when d≥3. Cheng and Sitharam strengthened Maxwell's condition by showing that every maximal d-sparse subgraph of G should have d|X|−(d+12) edges when d=3. We extend their result to all d≤11