XP tractability on forests for arbitrary patterns

Determine whether Pattern Avoidance parameterized by the pattern size n_P belongs to XP on forests, including the remaining case of patterns with a single forced edge that are not left-separated.

Background

The paper proves that Pattern Avoidance is polynomial-time solvable on forests for every constant-size non-mixed pattern. Mixed patterns, which have exactly one forced edge and are neither left-separated nor right-separated, are not covered by this algorithmic result. The authors establish NP-hardness for the related problem in which the first and last vertices of the ordering are prescribed, but this does not directly settle the unrestricted problem on forests.

Consequently, the authors leave open whether the XP result with respect to n_P extends to all patterns on forests, particularly the remaining non-left-separated case.

References

Is Pattern Avoidance in \XP\ w.r.t. $n_P$ on forests? The results in \Cref{sec:to:poly-trees} answer this affirmatively for all but a single case---however, the lower bound provided in Theorem~\ref{thm:one-sided-hard} can be seen as a strong indication of intractability for the remaining non-left separated case (cf.~Definition~\ref{def:poly-trees-preprocessing-left-separated}).

— The (Parameterized) Complexity of Ordering a Graph While Avoiding a Forbidden Pattern  (2608.30667 - Depian et al., 31 Aug 2026) in Section 6, Section 6.5, and Section 7 (Concluding Remarks)