Let $\mathcal{D}$ be a set of $n$ disks in the plane. The disk graph
$G_\mathcal{D}$ for $\mathcal{D}$ is the undirected graph with vertex set
$\mathcal{D}$ in which two disks are joined by an edge if and only if they
intersect. The directed transmission graph $G^{\rightarrow}_\mathcal{D}$ for
$\mathcal{D}$ is the directed graph with vertex set $\mathcal{D}$ in which
there is an edge from a disk $D_1 \in \mathcal{D}$ to a disk $D_2 \in
\mathcal{D}$ if and only if $D_1$ contains the center of $D_2$.
Given $\mathcal{D}$ and two non-intersecting disks $s, t \in \mathcal{D}$, we
show that a minimum $s$-$t$ vertex cut in $G_\mathcal{D}$ or in
$G^{\rightarrow}_\mathcal{D}$ can be found in $O(n^{3/2}\text{polylog} n)$
expected time. To obtain our result, we combine an algorithm for the maximum
flow problem in general graphs with dynamic geometric data structures to
manipulate the disks.
As an application, we consider the barrier resilience problem in a
rectangular domain. In this problem, we have a vertical strip $S$ bounded by
two vertical lines, $L_\ell$ and $L_r$, and a collection $\mathcal{D}$ of
disks. Let $a$ be a point in $S$ above all disks of $\mathcal{D}$, and let $b$
a point in $S$ below all disks of $\mathcal{D}$. The task is to find a curve
from $a$ to $b$ that lies in $S$ and that intersects as few disks of
$\mathcal{D}$ as possible. Using our improved algorithm for minimum cuts in
disk graphs, we can solve the barrier resilience problem in
$O(n^{3/2}\text{polylog} n)$ expected time.Comment: 11 pages, 4 figure