  Given a graph $G=(V,E)$, two vertices $s,t\in V$, and two integers $k,\ell$,
the Short Secluded Path problem is to find a simple $s$-$t$-path with at most
$k$ vertices and $\ell$ neighbors. We study the parameterized complexity of the
problem with respect to four structural graph parameters: the vertex cover
number, treewidth, feedback vertex number, and feedback edge number. In
particular, we completely settle the question of the existence of problem
kernels with size polynomial in these parameters and their combinations with
$k$ and $\ell$. We also obtain a $2^{O(w)}\cdot \ell^2\cdot n$-time algorithm
for graphs of treewidth $w$, which yields subexponential-time algorithms in
several graph classes.
