  In a proximity region graph ${\cal G}$ in $\mathbb{R}^d$, two distinct points
$x,y$ of a point process $\mu$ are connected when the 'forbidden region'
$S(x,y)$ these points determine has empty intersection with $\mu$. The Gabriel
graph, where $S(x,y)$ is the open disc with diameter the line segment
connecting $x$ and $y$, is one canonical example. When $\mu$ is a Poisson or
binomial process, under broad conditions on the regions $S(x,y)$, bounds on the
Kolmogorov and Wasserstein distances to the normal are produced for functionals
of ${\cal G}$, including the total number of edges and the total length.
Variance lower bounds, not requiring strong stabilization, are also proven to
hold for a class of such functionals.
