Fixed-parameter tractability of Path Cover parameterized by treewidth

Determine whether the Path Cover problem is fixed-parameter tractable when parameterized by treewidth alone.

Background

Path Cover asks whether the vertices of a graph can be covered by a specified number of paths. The paper contrasts this problem with Isometric Path Cover and notes that Path Cover is fixed-parameter tractable for the combined parameters of treewidth and solution size via Courcelle’s theorem.

The complexity of Path Cover under treewidth alone remains unresolved: the known results provide only an XP-type perspective, while fixed-parameter tractability has not been established or ruled out. This is a related open problem explicitly identified in the paper’s discussion of prior work.