Parameterized complexity by solution size of Isometric Path Cover and Isometric Path Partition

Determine whether Isometric Path Cover and Isometric Path Partition are fixed-parameter tractable or W[1]-hard when parameterized by the solution size k.

Background

The paper establishes several parameterized complexity results for Isometric Path Cover with respect to pathwidth, treewidth, and diameter, but does not resolve its complexity when the parameter is the number k of paths in the cover. Isometric Path Partition, the vertex-disjoint variant, is likewise unresolved under this parameterization.

Both problems are known to belong to XP for solution size k. The authors identify determining whether either problem is FPT or W[1]-hard under this parameterization as a principal unresolved direction for future research.

References

First, we wonder whether Isometric Path Cover and Isometric Path Partition are \FPT\ or \W[1]-hard when parameterized by the solution size $k$. {Both problem are known to be in \XP.} Arguably, these are the most important open questions on these problems from the perspective of parameterized complexity.

— Parameterized Complexity of Isometric Path Cover: Treewidth and Diameter  (2610.22010 - Chakraborty et al., 15 Oct 2026) in Section 6, Conclusion