  In the directed detour problem one is given a digraph $G$ and a pair of
vertices $s$ and~$t$, and the task is to decide whether there is a directed
simple path from $s$ to $t$ in $G$ whose length is larger than
$\mathsf{dist}_{G}(s,t)$. The more general parameterized variant, directed long
detour, asks for a simple $s$-to-$t$ path of length at least
$\mathsf{dist}_{G}(s,t)+k$, for a given parameter $k$. Surprisingly, it is
still unknown whether directed detour is polynomial-time solvable on general
digraphs. However, for planar digraphs, Wu and Wang~[Networks, '15] proposed an
$\mathcal{O}(n^3)$-time algorithm for directed detour, while Fomin et
al.~[STACS 2022] gave a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$-time fpt
algorithm for directed long detour. The algorithm of Wu and Wang relies on a
nontrivial analysis of how short detours may look like in a plane embedding,
while the algorithm of Fomin et al.~is based on a reduction to the
${\S}$-disjoint paths problem on planar digraphs. This latter problem is
solvable in polynomial time using the algebraic machinery of
Schrijver~[SIAM~J.~Comp.,~'94], but the degree of the obtained polynomial
factor is huge.
  In this paper we propose two simple algorithms: we show how to solve, in
planar digraphs, directed detour in time $\mathcal{O}(n^2)$ and directed long
detour in time $2^{\mathcal{O}(k)}\cdot n^4 \log n$. In both cases, the idea is
to reduce to the $2$-disjoint paths problem in a planar digraph, and to observe
that the obtained instances of this problem have a certain topological
structure that makes them amenable to a direct greedy strategy.
