We study two “above guarantee” versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an $(s,t)$-path of length at least ${\mathrm{dist}}_G(s,t)+k$. Bezáková et al. [7] proved that on undirected graphs the problem is fixed-parameter tractable ($\mathsf{FPT}$). Our first main result establishes a connection between Longest Detour on directed graphs and 3- Disjoint Paths on directed graphs. Using these new insights, we design a $2^{\mathcal{O}(k)}\cdot n^{\mathcal{O}(1)}$ time algorithm for the problem on directed planar graphs. Furthermore, the new approach yields a significantly faster $\mathsf{FPT}$ algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path above Diameter, the task is to decide whether the graph has a path of length at least $\mathrm{diam}(G)+k$. We obtain dichotomy results about Longest Path above Diameter on undirected and directed graphs.

我们研究了无向图和有向图上经典最长路径问题的两个“保证以上”版本，并获得以下结果。在我们研究的最长路径的第一个变体中，称为最长绕行，任务是决定一个图是否有一个$(秒,吨)$-至少长度的路径${\mathrm{距离}}_G(秒,吨)+k$. Bezáková 等人。[7]证明在无向图上问题是固定参数可处理的（$\mathsf{FPT}$). 我们的第一个主要结果在有向图上的最长绕行和有向图上的3 条不相交路径之间建立了联系。利用这些新见解，我们设计了一个${\mathrm{2个}}^{\mathcal{欧}(k)}\cdot n^{\mathcal{欧}(\mathrm{1个})}$有向平面图问题的时间算法。此外，新方法产生的速度明显更快$\mathsf{FPT}$无向图上的算法。在Longest Path的第二个变体中，即Longest Path above Diameter，任务是决定图是否至少有长度的路径$\mathrm{直径}(G)+k$. 我们在无向图和有向图上获得了直径以上最长路径的二分法结果。