Pattern Avoidance parameterized by treedepth and pattern size

Determine the complexity of Pattern Avoidance parameterized by the sum of the pattern size n_P and the treedepth of the input graph G, beginning with the special case of rainbow patterns corresponding to Queue Number.

Background

The paper proves that Pattern Avoidance is W[1]-hard when parameterized by the treedepth of G, even for patterns with one forced edge. It also notes that a fixed-parameter algorithm using treedepth together with n_P would have significant consequences for Queue Number, because rainbow patterns encode queue layouts.

The full complexity of the combined parameterization by n_P plus the treedepth of G is therefore unresolved. The authors identify determining the corresponding complexity for rainbow patterns, or equivalently Queue Number parameterized by treedepth, as a natural first step.