Parameterized complexity of Terminal Monitoring Set for structural parameters (pathwidth, treewidth, feedback vertex number, and path-deletion)

Determine whether the Terminal Monitoring Set problem is fixed-parameter tractable or W[1]-hard when parameterized by each of the following graph parameters: pathwidth, treewidth, feedback vertex number, and the vertex deletion number to a disjoint union of paths.

Background

Terminal Monitoring Set (TMS) asks for a minimum-size vertex set that intersects a shortest path between every specified pair of terminals in an undirected graph. The paper establishes NP-hardness and W[2]-hardness with respect to solution size, while giving fixed-parameter tractable (FPT) algorithms under several structural parameterizations (e.g., solution size plus distance to cluster, solution size plus neighborhood diversity, and feedback edge number). A weighted variant is shown FPT by vertex cover number.

Despite these advances, the parameterized complexity of TMS under several classic and widely studied structural parameters remains unsettled. In particular, the authors explicitly leave open whether TMS is FPT or hard when parameterized by pathwidth, treewidth, feedback vertex number, and the vertex deletion number to a disjoint union of paths.