Given a set $P$ of $n$ points in the plane and a collection of disks centered at these points, the disk graph $G(P)$ has vertex set $P$, with an edge between two vertices if their corresponding disks intersect. We study the dominating set problem in $G(P)$ under the special case where the points of $P$ are in convex position. The problem is NP-hard in general disk graphs. Under the convex position assumption, however, we present the first polynomial-time algorithm for the problem. Specifically, we design an $O(k^2 n \log^2 n)$-time algorithm, where $k$ denotes the size of a minimum dominating set. For the weighted version, in which each disk has an associated weight and the goal is to compute a dominating set of minimum total weight, we obtain an $O(n^5 \log^2 n)$-time algorithm.

翻译：给定平面上一个包含$n$个点的集合$P$以及一组以这些点为中心的圆盘，圆盘图$G(P)$以$P$为顶点集，若两个顶点对应的圆盘相交，则它们之间存在一条边。我们研究$G(P)$中支配集问题的特殊情况，即$P$中的点处于凸位置。该问题在一般圆盘图中是NP难的。然而，在凸位置假设下，我们提出了该问题的首个多项式时间算法。具体而言，我们设计了一个时间复杂度为$O(k^2 n \log^2 n)$的算法，其中$k$表示最小支配集的大小。对于加权版本，其中每个圆盘具有关联权重，目标是计算总权重最小的支配集，我们得到了一个时间复杂度为$O(n^5 \log^2 n)$的算法。