Before revision:  The directed transmission graph G^{\rightarrow}_D for D is the directed graph with vertex set D in which there is an edge from a disk D_1 \in D to a disk D_2 \in D if and only if D_1 contains the center of D_2. Given D and two non-intersecting disks s, t \in D, we show that a minimum s-t vertex cut in G_D or in G^{\rightarrow}_\mathcal{D} can be found in O(n^{3/2}   n) expected time.
After revision:  The directed transmission graph G^{\rightarrow}_D for D is the directed graph with vertex set D in which there is an edge from a disk D_1 \in D to a disk D_2 \in D if and only if D_1 contains the center of D_2. Given D and two non-intersecting disks s, t \in D, we show that a minimum s-t vertex cut in G_D or in G^{\rightarrow}_\mathcal{D} can be found in O(n^{3/2}  \text{polylog n) expected time.