  Given a hypergraph $H$, the Planar Support problem asks whether there is a
planar graph $G$ on the same vertex set as $H$ such that each hyperedge induces
a connected subgraph of $G$. Planar Support is motivated by applications in
graph drawing and data visualization. We show that Planar Support is
fixed-parameter tractable when parameterized by the number of hyperedges in the
input hypergraph and the outerplanarity number of the sought planar graph. To
this end, we develop novel structural results for $r$-outerplanar triangulated
disks, showing that they admit sequences of separators with structural
properties enabling data reduction. This allows us to obtain a problem kernel
for Planar Support, thus showing its fixed-parameter tractability.
