  Given a directed graph $G$ and a pair of nodes $s$ and $t$, an \emph{$s$-$t$
bridge} of $G$ is an edge whose removal breaks all $s$-$t$ paths of $G$ (and
thus appears in all $s$-$t$ paths). Computing all $s$-$t$ bridges of $G$ is a
basic graph problem, solvable in linear time.
  In this paper, we consider a natural generalisation of this problem, with the
notion of "safety" from bioinformatics. We say that a walk $W$ is \emph{safe}
with respect to a set $\mathcal{W}$ of $s$-$t$ walks, if $W$ is a subwalk of
all walks in $\mathcal{W}$. We start by considering the maximal safe walks when
$\mathcal{W}$ consists of: all $s$-$t$ paths, all $s$-$t$ trails, or all
$s$-$t$ walks of $G$. We show that the first two problems are immediate
linear-time generalisations of finding all $s$-$t$ bridges, while the third
problem is more involved. In particular, we show that there exists a compact
representation computable in linear time, that allows outputting all maximal
safe walks in time linear in their length.
  We further generalise these problems, by assuming that safety is defined only
with respect to a subset of \emph{visible} edges. Here we prove a dichotomy
between the $s$-$t$ paths and $s$-$t$ trails cases, and the $s$-$t$ walks case:
the former two are NP-hard, while the latter is solvable with the same
complexity as when all edges are visible. We also show that the same complexity
results hold for the analogous generalisations of \emph{$s$-$t$ articulation
points} (nodes appearing in all $s$-$t$ paths).
  We thus obtain the best possible results for natural "safety"-generalisations
of these two fundamental graph problems. Moreover, our algorithms are simple
and do not employ any complex data structures, making them ideal for use in
practice.
