  For a connected graph $G = (V, E)$ and $s, t \in V$, a non-separating $s$-$t$
path is a path $P$ between $s$ and $t$ such that the set of vertices of $P$
does not separate $G$, that is, $G - V(P)$ is connected. An $s$-$t$ path is
non-disconnecting if $G - E(P)$ is connected. The problems of finding shortest
non-separating and non-disconnecting paths are both known to be NP-hard. In
this paper, we consider the problems from the viewpoint of parameterized
complexity. We show that the problem of finding a non-separating $s$-$t$ path
of length at most $k$ is W[1]-hard parameterized by $k$, while the
non-disconnecting counterpart is fixed-parameter tractable parameterized by
$k$. We also consider the shortest non-separating path problem on several
classes of graphs and show that this problem is NP-hard even on bipartite
graphs, split graphs, and planar graphs. As for positive results, the shortest
non-separating path problem is fixed-parameter tractable parameterized by $k$
on planar graphs and polynomial-time solvable on chordal graphs if $k$ is the
shortest path distance between $s$ and $t$.
