We show that if the edges or vertices of an undirected graph $G$ can be covered by $k$ shortest paths, then the pathwidth of $G$ is upper-bounded by a single-exponential function of $k$. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph $G$ and a set of $k$ pairs of vertices called terminals, asks whether $G$ can be covered by $k$ shortest paths, each joining a pair of terminals) is FPT with respect to the number of terminals. The same holds for the similar problem Strong Geodetic Set with Terminals (which, given a graph $G$ and a set of $k$ terminals, asks whether there exist $\binom{k}{2}$ shortest paths covering $G$, each joining a distinct pair of terminals). Moreover, this implies that the related problems Isometric Path Cover and Strong Geodetic Set (defined similarly but where the set of terminals is not part of the input) are in XP with respect to parameter $k$.

翻译：我们证明：如果无向图$G$的边或顶点可被$k$条最短路径覆盖，则$G$的路径宽度被一个关于$k$的单指数函数所上界约束。作为推论，我们证明了带端点的等距路径覆盖问题（给定图$G$和$k$对称为端点的顶点，询问$G$是否可被$k$条分别连接一对端点的最短路径覆盖）关于端点数量是固定参数可解的。这一结论同样适用于类似的带端点的强测地集问题（给定图$G$和$k$个端点，询问是否存在$\binom{k}{2}$条覆盖$G$的最短路径，每条路径连接一对不同的端点）。此外，这意味着相关问题——等距路径覆盖和强测地集（定义类似但端点集不作为输入的一部分）——关于参数$k$属于XP类。