Parameterized Complexity of Isometric Path Cover: Treewidth and Diameter

Abstract: We study the fixed-parameter tractability of the \textsc{Isometric Path Cover} problem where, given an integer $k$ and a simple, undirected, and unweighted graph $G$, the objective is to decide whether $G$ admits a set of $k$ isometric paths (i.e., shortest paths between their endpoints) whose union is the entire vertex set of $G$. We show that \textsc{Isometric Path Cover} is NP-hard on graphs of pathwidth (and thus treewidth) at most 287. This is an interesting contrast with the closely related problems \textsc{Isometric Path Partition} and \textsc{Path Cover}, which are both known to be XP for treewidth, see [Chakraborty et al., WG 2026] and [Foucaud et al., TCS 2026]. On the positive side, we show, by a suitable dynamic programming scheme, that \textsc{Isometric Path Cover} admits an XP algorithm w.r.t.~the parameter combination of treewidth and diameter, running in time $n<sup>{\diam<sup>{O(\tw)}}$ where $\diam$, $\tw$, and $n$ denote the diameter, the treewidth, and the number of vertices of the input graph, respectively. However, we also prove that \textsc{Isometric Path Cover} becomes W[1]-hard for pathwidth on graphs of diameter~7, showing that an XP algorithm is the best one can hope for w.r.t.~this parameterization, under standard complexity theory assumptions.