  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 yields a
significantly faster 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 diam(G)+k
(diam(G) denotes the length of a longest shortest path in a graph G). We obtain
dichotomy results about Longest Path Above Diameter on undirected and directed
graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete
even for k=1. However, if the input undirected graph is 2-connected, then the
problem is FPT. On the other hand, for 2-connected directed graphs, we show
that Longest Path Above Diameter is solvable in polynomial time for each
k\in{1,\dots, 4} and is NP-complete for every k\geq 5. The parameterized
complexity of Longest Path Above Diameter on general directed graphs remains an
interesting open problem.
