Detours in Directed Graphs
Fedor V. Fomin, Petr A. Golovach, William Lochet, Danil Sagunov,, Kirill Simonov, Saket Saurabh

TL;DR
This paper investigates parameterized variants of the Longest Path problem, establishing fixed-parameter tractability results for directed and undirected graphs and providing complexity classifications for path length problems relative to graph diameter.
Contribution
It introduces new algorithmic results for Longest Detour on directed graphs and complexity dichotomies for Longest Path Above Diameter, advancing understanding of these problems.
Findings
FPT algorithm for Longest Detour on directed planar graphs
FPT algorithm for Longest Detour on undirected graphs
NP-completeness and polynomial cases for Longest Path Above Diameter
Abstract
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 dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bez\'akov\'a et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O (k)} n. Further, they left the parameterized complexity of the problem on directed graphs open. 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^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach…
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