  We consider the situation where one is given a set S of points in the plane
and a collection D of unit disks embedded in the plane. We show that finding a
minimum cardinality subset of D such that any path between any two points in S
is intersected by at least one disk is NP-complete. This settles an open
problem raised by Matt Gibson et al[1]. Using a similar reduction, we show that
finding a minimum cardinality subset D' of D such that R^2 - (D - D') consists
of a single connected region is also NP-complete. Lastly, we show that the
Multiterminal Cut Problem remains NP-complete when restricted to unit disk
graphs.
