Let $G=(V,E)$ be a geometric graph, a graph embedded in the plane whose edge lengths are
the Euclidean distance between its endpoint vertices.
Say that a set of vertices $D \subseteq V$ is a *geometric dominating set*
if for every $v \in V$, the shortest path in the graph
from $v$ to some vertex in $D$ is at most length $1$.

For example, a pentagonal wheel with spokes of length $1$ can be dominated with one vertex, but if the shortest edge length is greater than $1$, then $D$ must equal $V$:

The traditional dominating set problem has been known to be NP-complete since at least Garey & Johnson's 1979 book. My question is:

. Is finding a smallest geometric dominating set for a given geometric graph also NP-complete?Q

Conceivably it is not intractable, if the geometric structure can be exploited. Perhaps plane graphs $G$ would especially allow that exploitation. If anyone knows of references, I would appreciate pointers. Thanks!