On Graphs Coverable by k Shortest Paths.

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 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 \emph{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, each joining a distinct pair of terminals such that these paths cover $G$). 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 $k$.