On Graphs Coverable by \({k}\) Shortest Paths
Maël Dumas, Florent Foucaud, Anthony Perez, Ioan Todinca · SIAM Journal on Discrete Mathematics · 2024
Abstract. We show that if the edges or vertices of an undirected graph [Formula: see text] can be covered by [Formula: see text] shortest paths, then the pathwidth of [Formula: see text] is upper-bounded by a single-exponential function of [Formula: see text]. As a corollary, we prove that the problem Isometric Path Cover with Terminals (which, given a graph [Formula: see text] and a set of [Formula: see text] pairs of vertices called terminals, asks whether [Formula: see text] can be covered by [Formula: see text] 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 [Formula: see text] and a set of [Formula: see text] terminals, asks whether there exist [Formula: see text] shortest paths covering [Formula: see text], 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 [Formula: see text].