On graphs coverable by k shortest paths

TL;DR

This paper establishes a bound on the pathwidth of graphs coverable by k shortest paths and proves fixed-parameter tractability for related path cover problems with respect to the number of terminals.

Contribution

It introduces a single-exponential bound on pathwidth for graphs covered by k shortest paths and shows FPT algorithms for the Isometric Path Cover with Terminals problem.

Findings

Pathwidth is bounded by a single-exponential function of k.

Isometric Path Cover with Terminals is FPT in k.

Related problems are in XP with respect to k.

Abstract

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 $k$.