  We consider intersection graphs of disks of radius $r$ in the hyperbolic
plane. Unlike the Euclidean setting, these graph classes are different for
different values of $r$, where very small $r$ corresponds to an
almost-Euclidean setting and $r \in \Omega(\log n)$ corresponds to a firmly
hyperbolic setting. We observe that larger values of $r$ create simpler graph
classes, at least in terms of separators and the computational complexity of
the \textsc{Independent Set} problem.
  First, we show that intersection graphs of disks of radius $r$ in the
hyperbolic plane can be separated with $\mathcal{O}((1+1/r)\log n)$ cliques in
a balanced manner. Our second structural insight concerns Delaunay complexes in
the hyperbolic plane and may be of independent interest. We show that for any
set $S$ of $n$ points with pairwise distance at least $2r$ in the hyperbolic
plane the corresponding Delaunay complex has outerplanarity
$1+\mathcal{O}(\frac{\log n}{r})$, which implies a similar bound on the
balanced separators and treewidth of such Delaunay complexes.
  Using this outerplanarity (and treewidth) bound we prove that
\textsc{Independent Set} can be solved in $n^{\mathcal{O}(1+\frac{\log n}{r})}$
time. The algorithm is based on dynamic programming on some unknown sphere cut
decomposition that is based on the solution. The resulting algorithm is a
far-reaching generalization of a result of Kisfaludi-Bak (SODA 2020), and it is
tight under the Exponential Time Hypothesis. In particular, \textsc{Independent
Set} is polynomial-time solvable in the firmly hyperbolic setting of $r\in
\Omega(\log n)$. Finally, in the case when the disks have ply (depth) at most
$\ell$, we give a PTAS for \textsc{Maximum Independent Set} that has only
quasi-polynomial dependence on $1/\varepsilon$ and $\ell$. Our PTAS is a
further generalization of our exact algorithm.
