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.

翻译：在直接偏向问题中, 给一个直径 $G$和一对直径 $4 美元和 美元, 任务在于决定一个指定的简单路径, 从美元到美元, 其长度大于$mathsf{dist ⁇ G} 美元。 更普遍的参数变量, 引导长路, 要求一个简单的 $- 美元到 美元, 至少是 maths@ divs} dial_ g} 美元 +k$, 给一个指定的参数 $4, 美元和 美元。 令人惊讶的是, 仍然不知道一个直径路径的直径路径是否大于$masf{distg} 美元。 然而, 平面的平面图, Wu 和 Wang~ [Networks, '15] 都直接提出一个 mathcal{O} 问题, 直径直径的直径直径的直径算算算算算算算出一个简单的 。 Formal- max 时间里, 直径直径解算算算算算算算算出一个2 mal_ tal_O} 时间。