Hyperbolic tilings are natural infinite planar graphs where each vertex has degree $q$ and each face has $p$ edges for some $\frac1p+\frac1q<\frac12$. We study the structure of shortest paths in such graphs. We show that given a set of $n$ terminals, we can compute a so-called isometric closure (closely related to the geodesic convex hull) of the terminals in near-linear time, using a classic geometric convex hull algorithm as a black box. We show that the size of the convex hull is $O(N)$ where $N$ is the total length of the paths to the terminals from a fixed origin. Furthermore, we prove that the geodesic convex hull of a set of $n$ terminals has treewidth only $\max(12,O(\log\frac{n}{p + q}))$, a bound independent of the distance of the points involved. As a consequence, we obtain algorithms for subset TSP and Steiner tree with running time $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$.

翻译：双曲铺砌是一类自然的无限平面图，其中每个顶点的度为 $q$，每个面有 $p$ 条边，满足 $\frac1p+\frac1q<\frac12$。我们研究了此类图中最短路径的结构。我们证明，给定一组 $n$ 个终端点，可以在近线性时间内计算这些终端点的等距闭包（与测地凸包密切相关），其实现方式是将经典几何凸包算法作为黑盒使用。我们证明凸包的大小为 $O(N)$，其中 $N$ 是从固定原点到所有终端点的路径总长度。此外，我们证明了 $n$ 个终端点的测地凸包的树宽仅为 $\max(12,O(\log\frac{n}{p + q}))$，该界限与所涉及点的距离无关。基于此，我们得到了子集旅行商问题（subset TSP）和斯坦纳树问题的算法，其运行时间为 $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$。