  A graph $G=(V,E)$ is $d$-sparse if each subset $X\subseteq V$ with $|X|\geq
d$ induces at most $d|X|-{{d+1}\choose{2}}$ 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 ${\mathbb R}^d$ is that $G$ should have a
$d$-sparse subgraph with $d|X|-{{d+1}\choose{2}}$ edges. This necessary
condition is also sufficient when $d=1,2$ but not when $d\geq 3$. Cheng and
Sitharam strengthened Maxwell's condition by showing that every maximal
$d$-sparse subgraph of $G$ should have $d|X|-{{d+1}\choose{2}}$ edges when
$d=3$. We extend their result to all $d\leq 11$.
