The hyperbolicity of a graph, informally, measures how close a graph is (metrically) to a tree. Hence, it is intuitively similar to treewidth, but the measures are formally incomparable. Motivated by the broad study of algorithms and separators on planar graphs and their relation to treewidth, we initiate the study of planar graphs of bounded hyperbolicity. Our main technical contribution is a novel balanced separator theorem for planar $\delta$-hyperbolic graphs that is substantially stronger than the classic planar separator theorem. For any fixed $\delta \geq 0$, we can find balanced separator that induces either a single geodesic (shortest) path or a single geodesic cycle in the graph. An important advantage of our separator is that the union of our separator (vertex set $Z$) with any subset of the connected components of $G - Z$ induces again a planar $\delta$-hyperbolic graph, which would not be guaranteed with an arbitrary separator. Our construction runs in near-linear time and guarantees that size of separator is $\mathrm{poly}(\delta) \cdot \log n$. As an application of our separator theorem and its strong properties, we obtain two novel approximation schemes on planar $\delta$-hyperbolic graphs. We prove that Maximum Independent Set and the Traveling Salesperson problem have a near-linear time FPTAS for any constant $\delta$, running in $n\, \mathrm{polylog}(n) \cdot 2^{\mathcal{O}(\delta^2)} \cdot \varepsilon^{-\mathcal{O}(\delta)}$ time. We also show that our approximation scheme for Maximum Independent Set has essentially the best possible running time under the Exponential Time Hypothesis (ETH). This immediately follows from our third contribution: we prove that Maximum Independent Set has no $n^{o(\delta)}$-time algorithm on planar $\delta$-hyperbolic graphs, unless ETH fails.

翻译：图的双曲性非正式地度量一个图在度量上接近树的程度。因此，它在直觉上与树宽相似，但这两个度量在形式上是不可比较的。受平面图上的算法与分隔符及其与树宽关系的广泛研究的启发，我们开创了对有界双曲性平面图的研究。我们的主要技术贡献是针对平面 δ-双曲图的一个新颖的平衡分隔定理，该定理显著强于经典的平面分隔定理。对于任意固定的 δ ≥ 0，我们可以找到一个平衡分隔符，该分隔符在图内诱导出一条单一测地线（最短路径）或一个单一测地环。我们分隔符的一个重要优势是，该分隔符（顶点集 Z）与 G - Z 的任意连通分量子集的并集再次诱导出一个平面 δ-双曲图，而这在使用任意分隔符时是无法保证的。我们的构造以近线性时间运行，并保证分隔符的大小为 poly(δ)·log n。作为我们的分隔定理及其强大性质的一个应用，我们在平面 δ-双曲图上获得了两种新颖的近似方案。我们证明了最大独立集问题和旅行商问题对任意常数 δ 都有一个近线性时间的 FPTAS，其运行时间为 n polylog(n)·2^{O(δ^2)}·ε^{-O(δ)}。我们还表明，在指数时间假设（ETH）下，我们的最大独立集近似方案本质上具有最佳可能的运行时间。这直接源于我们的第三个贡献：我们证明了除非 ETH 失败，否则平面 δ-双曲图上的最大独立集问题不存在 n^{o(δ)} 时间算法。