Faithful-emulator generalization to minor-free digraphs

Generalize the faithful reachability-emulator construction from planar digraphs to K_h-minor-free digraphs for fixed h, including a near-linear construction based on balanced separators formed by O(1) directed paths.

Background

The paper gives a truly sublinear fully dynamic reachability structure for K_h-minor-free digraphs, but its stronger planar results do not transfer directly. One obstacle is the faithful-emulator construction, whose proof relies on balanced separators consisting of a constant number of directed paths. The authors identify the existence and near-linear computability of suitable separators in minor-free digraphs as unresolved.

References

Second, it is unclear whether the construction from~\Cref{sec:faithful} can be generalized to $K_h$-minor-free digraphs due to the need of finding balanced separators formed by $O(1)$ dipaths (\Cref{lem:faithful:balanced}).

— Breaking Two Barriers for Dynamic Planar Reachability  (2610.13473 - Karczmarz et al., 9 Oct 2026) in Section 7, Conclusion, paragraph “Minor-free digraphs”

Since this construction requires full access to the Robertson-Seymour decomposition of a graph, improving the running time to near-linear (or close-to-linear) seems challenging.

— Breaking Two Barriers for Dynamic Planar Reachability  (2610.13473 - Karczmarz et al., 9 Oct 2026) in Section 7, Conclusion, paragraph “Minor-free digraphs”