Bounds to the normal for proximity region graphs

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. Under broad conditions on the process $\mu$ and 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 total length.