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.

翻译：给定一个无向图 $G=(V,E)$ 和一个整数 $\ell$，偏心率最短路径问题（ESP）要求找到一条最短路径 $P$，使得对于每个顶点 $v\in V(G)$，存在一个顶点 $w\in P$ 满足 $d_G(v,w)\leq \ell$，其中 $d_G(v,w)$ 表示 $v$ 和 $w$ 在图 $G$ 中的距离。Dragan 和 Leitert [Theor. Comput. Sci. 2017] 证明了该问题的优化版本（即寻找 ESP 问题的最小 $\ell$）即使在最大度为 3 的平面二部图上也是 NP-难的。他们还证明了当以 $\ell$ 为参数时，ESP 是 W[2]-难的。从正面来看，Kučera 和 Suchý [IWOCA 2021] 表明，当以模宽度、团顶点删除集、最大叶子数或组合参数（不交路径删除集和 $\ell$）为参数时，该问题具有固定参数可解（FPT）性质。上述论文提出了一个开放性问题：ESP 是否在以不交路径删除集或反馈顶点集为参数时是 FPT 的？我们部分回答了这些问题，并得到以下结果：- ESP 在以不交路径删除集、分裂顶点删除集或组合参数（反馈顶点集和图的偏心率）为参数时是 FPT 的。- 我们设计了一个以反馈顶点集数为参数的 $(1+\epsilon)$-因子 FPT 近似算法。- ESP 在以弦顶点删除集为参数时是 W[2]-难的。