  In this paper we present an efficient reachability oracle under single-edge
or single-vertex failures for planar directed graphs. Specifically, we show
that a planar digraph $G$ can be preprocessed in $O(n\log^2{n}/\log\log{n})$
time, producing an $O(n\log{n})$-space data structure that can answer in
$O(\log{n})$ time whether $u$ can reach $v$ in $G$ if the vertex $x$ (the
edge~$f$) is removed from $G$, for any query vertices $u,v$ and failed vertex
$x$ (failed edge $f$). To the best of our knowledge, this is the first data
structure for planar directed graphs with nearly optimal preprocessing time
that answers all-pairs queries under any kind of failures in polylogarithmic
time.
  We also consider 2-reachability problems, where we are given a planar digraph
$G$ and we wish to determine if there are two vertex-disjoint (edge-disjoint)
paths from $u$ to $v$, for query vertices $u,v$. In this setting we provide a
nearly optimal 2-reachability oracle, which is the existential variant of the
reachability oracle under single failures, with the following bounds. We can
construct in $O(n\log^{O(1)}{n})$ time an $O(n\log^{3+o(1)}{n})$-space data
structure that can check in $O(\log^{2+o(1)}{n})$ time for any query vertices
$u,v$ whether $v$ is 2-reachable from $u$, or otherwise find some separating
vertex (edge) $x$ lying on all paths from $u$ to $v$ in $G$.
  To obtain our results, we follow the general recursive approach of Thorup for
reachability in planar graphs [J.~ACM~'04] and we present new data structures
which generalize dominator trees and previous data structures for
strong-connectivity under failures [Georgiadis et al., SODA~'17]. Our new data
structures work also for general digraphs and may be of independent interest.
