Strictness of the W-Hierarchy Inclusions

Prove that the inclusions text{FPT} subseteq W[1] subseteq W[2] dots subseteq W[P] are strict.

Background

The paper recalls the standard parameterized-complexity conjecture that the levels of the W-hierarchy do not collapse. This conjecture underlies the interpretation of W[t]-hardness results: if the inclusions are strict, then hardness for a class above FPT provides evidence that a problem is not fixed-parameter tractable.