Let $d$ be a (well-behaved) shortest-path metric defined on a path-connected subset of $\mathbb{R}^2$ and let $\mathcal{D}=\{D_1,\ldots,D_n\}$ be a set of geodesic disks with respect to the metric $d$. We prove that $\mathcal{G}^{\times}(\mathcal{D})$, the intersection graph of the disks in $\mathcal{D}$, has a clique-based separator consisting of $O(n^{3/4+\varepsilon})$ cliques. This significantly extends the class of objects whose intersection graphs have small clique-based separators. Our clique-based separator yields an algorithm for $q$-COLORING that runs in time $2^{O(n^{3/4+\varepsilon})}$, assuming the boundaries of the disks $D_i$ can be computed in polynomial time. We also use our clique-based separator to obtain a simple, efficient, and almost exact distance oracle for intersection graphs of geodesic disks. Our distance oracle uses $O(n^{7/4+\varepsilon})$ storage and can report the hop distance between any two nodes in $\mathcal{G}^{\times}(\mathcal{D})$ in $O(n^{3/4+\varepsilon})$ time, up to an additive error of one. So far, distance oracles with an additive error of one that use subquadratic storage and sublinear query time were not known for such general graph classes.

翻译：设 $d$ 为定义在 $\mathbb{R}^2$ 的路径连通子集上的（良好行为的）最短路径度量，并设 $\mathcal{D}=\{D_1,\ldots,D_n\}$ 为关于度量 $d$ 的一组测地圆盘。我们证明 $\mathcal{G}^{\times}(\mathcal{D})$（即 $\mathcal{D}$ 中圆盘的交图）存在一个由 $O(n^{3/4+\varepsilon})$ 个团组成的基于团的分离器。这显著扩展了其交图具有小规模基于团分离器的对象类别。我们的基于团的分离器为 $q$ 染色问题提供了一种时间复杂度为 $2^{O(n^{3/4+\varepsilon})}$ 的算法，前提是圆盘 $D_i$ 的边界可以在多项式时间内计算。我们还利用该基于团的分离器，为测地圆盘交图构造了一个简单、高效且几乎精确的距离预言。该距离预言使用 $O(n^{7/4+\varepsilon})$ 的存储空间，并能在 $O(n^{3/4+\varepsilon})$ 时间内报告 $\mathcal{G}^{\times}(\mathcal{D})$ 中任意两点间的跳数距离，且附加误差不超过一。迄今为止，对于如此广泛的图类，尚未有已知的具有次二次存储和次线性查询时间且附加误差为一的距离预言。