A graph G=(V,E) is d-sparse if each subset XβV with β£Xβ£β₯d induces at most dβ£Xβ£β(2d+1β) 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 dβ£Xβ£β(2d+1β) edges. This necessary
condition is also sufficient when d=1,2 but not when dβ₯3. Cheng and
Sitharam strengthened Maxwell's condition by showing that every maximal
d-sparse subgraph of G should have dβ£Xβ£β(2d+1β) edges when
d=3. We extend their result to all dβ€11.Comment: There was an error in the proof of Theorem 3.3(b) in version 1 of
this paper. A weaker statement was proved in version 2 and then used to
derive the main result Theorem 4.1 when dβ€5. The proof technique was
subsequently refined in collaboration with Hakan Guler to extend this result
to all dβ€11 in Theorem 3.3 of version