Fixed-parameter tractability with respect to the number of vertices

Determine whether temporal agony is fixed-parameter tractable when parameterized by the number n of vertices, including the case of three ranks.

Background

The paper gives an XP algorithm parameterized by the number n of vertices and an FPT algorithm for the combined parameter n+ℓ, where ℓ is the number of allowed rank changes per vertex. It does not establish fixed-parameter tractability for n alone and explicitly notes that this remains unresolved even when the number of ranks is restricted to three.