On isolating points using unit disks

Given a set of points in the plane and a set of disks (that we think of as wireless sensors) which separate the points, we consider the problem of selecting a minimum subset of the disks such that any path between any pair of points is intersected by at least one of the selected disks. We present a $(9 + \epsilon)$-approximation algorithm for this problem and show that it is NP-complete even if all disks have unit radius and no disk contains any points. Using a similar hardness reduction, we further show that the Multiterminal Cut problem remains NP-complete on unit disk graphs. Lastly, we prove that removing a minimum subset of a collection of unit disks, such that the plane minus the  arrangement of the remaining disks consists of a single connected region is also NP-complete.