In the directed detour problem, given a directed graph $G$ and a pair of vertices $s$ and ~$t$, the task is whether there is a directed simple path from $s$ to $t$ in $. It’s about deciding what G$ with length greater than $\mathsf{dist}_{G}(s,t)$. A more general parameterized variant, the directed long detour, is given the parameter $k$. Surprisingly, it is not yet known whether directed detours can be solved in general directed graphs in polynomial time. However, for planar directed graphs, Wu and Wang~[Networks, ’15] proposed a $\mathcal{O}(n^3)$-time algorithm for directed detours, Fomin et al.[STACS 2022] $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$-time gave the fpt algorithm to the specified long detour. Wu and Wang’s algorithm relies on a significant analysis of what short detours look like in planar embeddings, whereas Fomin et al. based on a reduction to Planar directed graph. This latter problem can be solved in polynomial time using the Schrijver~ algebraic machine.[SIAM~J.~Comp.,~’94]but the degree of the resulting polynomial coefficients is huge.
In this paper, we propose two simple algorithms. In a planar directed graph, the directed detour time $\mathcal{O}(n^2)$ and the directed long detour time $2^{\mathcal{O}( k)}\cdot n^4 \log n$. Also, the idea is to reduce to the problem of $2$-disjoint paths in planar directed graphs and observe that the captured instance of this problem has a certain topological structure that follows a direct greedy strategy.
