  Given an undirected graph $G=(V,E)$ and an integer $\ell$, the Eccentricity
Shortest Path (ESP) asks to find a shortest path $P$ such that for every vertex
$v\in V(G)$, there is a vertex $w\in P$ such that $d_G(v,w)\leq \ell$, where
$d_G(v,w)$ represents the distance between $v$ and $w$ in $G$. Dragan and
Leitert [Theor. Comput. Sci. 2017] showed that the optimization version of this
problem, which asks to find the minimum $\ell$ for the ESP problem, is NP-hard
even on planar bipartite graphs with maximum degree 3. They also showed that
ESP is W[2]-hard when parameterized by $\ell$. On the positive side, Ku\v cera
and Such\'y [IWOCA 2021] showed that the problem exhibits fixed parameter
tractable (FPT) behavior when parameterized by modular width, cluster vertex
deletion set, maximum leaf number, or the combined parameters disjoint paths
deletion set and $\ell$. It was asked as an open question in the above paper,
if ESP is FPT parameterized by disjoint paths deletion set or feedback vertex
set. We answer these questions partially and obtain the following results: -
ESP is FPT when parameterized by disjoint paths deletion set, split vertex
deletion set or the combined parameters feedback vertex set and eccentricity of
the graph. - We design a $(1+\epsilon)$-factor FPT approximation algorithm when
parameterized by the feedback vertex set number. - ESP is W[2]-hard when
parameterized by the chordal vertex deletion set.
