  In a strongly connected graph $G = (V,E)$, a cut arc (also called strong
bridge) is an arc $e \in E$ whose removal makes the graph no longer strongly
connected. Equivalently, there exist $u,v \in V$, such that all $u$-$v$ walks
contain $e$. Cut arcs are a fundamental graph-theoretic notion, with countless
applications, especially in reachability problems.
  In this paper we initiate the study of cut paths, as a generalisation of cut
arcs, which we naturally define as those paths $P$ for which there exist $u,v
\in V$, such that all $u$-$v$ walks contain $P$ as subwalk. We first prove
various properties of cut paths and define their remainder structures, which we
use to present a simple $O(m)$-time verification algorithm for a cut path ($|V|
= n$, $|E| = m$).
  Secondly, we apply cut paths and their remainder structures to improve
several reachability problems from bioinformatics. A walk is called safe if it
is a subwalk of every node-covering closed walk of a strongly connected graph.
Multi-safety is defined analogously, by considering node-covering sets of
closed walks instead. We show that cut paths provide simple $O(m)$-time
algorithms verifying if a walk is safe or multi-safe. For multi-safety, we
present the first linear time algorithm, while for safety, we present a simple
algorithm where the state-of-the-art employed complex data structures. Finally
we show that the simultaneous computation of remainder structures of all
subwalks of a cut path can be performed in linear time. These properties yield
an $O(mn)$ algorithm outputting all maximal multi-safe walks, improving over
the state-of-the-art algorithm running in time $O(m^2+n^3)$.
  The results of this paper only scratch the surface in the study of cut paths,
and we believe a rich structure of a graph can be revealed, considering the
perspective of a path, instead of just an arc.
