  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.
