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.

翻译：设$\mathcal{D}$为平面上的$n$个圆盘构成的集合。圆盘图$G_\mathcal{D}$是以$\mathcal{D}$为顶点集的无向图，其中两个圆盘之间当且仅当它们相交时存在一条边。有向传输图$G^{\rightarrow}_\mathcal{D}$是以$\mathcal{D}$为顶点集的有向图，其中从圆盘$D_1 \in \mathcal{D}$到圆盘$D_2 \in \mathcal{D}$存在一条边当且仅当$D_1$包含$D_2$的圆心。给定$\mathcal{D}$及两个不相交的圆盘$s, t \in \mathcal{D}$，我们证明可以在$O(n^{3/2}\text{polylog} n)$期望时间内找到$G_\mathcal{D}$或$G^{\rightarrow}_\mathcal{D}$中的最小$s$-$t$顶点割。为得到这一结果，我们将一般图中的最大流算法与操纵圆盘的动态几何数据结构相结合。作为应用，我们考虑矩形区域中的屏障韧性问题。在该问题中，我们有一个由两条垂直线$L_\ell$和$L_r$界定的垂直带$S$，以及一个圆盘集合$\mathcal{D}$。设$a$为$S$中位于所有$\mathcal{D}$圆盘上方的点，$b$为$S$中位于所有$\mathcal{D}$圆盘下方的点。任务是从$a$到$b$找到一条位于$S$内且与$\mathcal{D}$中尽可能少圆盘相交的曲线。利用我们改进的圆盘图最小割算法，可以在$O(n^{3/2}\text{polylog} n)$期望时间内解决屏障韧性问题。