Generalizing the distance-d pattern-sparse framework to H-minor-free graphs

Determine whether the distance-d (d ≥ 1) pattern-sparse tree-decomposition framework that currently yields subexponential-time algorithms by ensuring each bag intersects Nd[Z] in O(√k polylog k) vertices for K3,h-minor-free graphs can be extended to all H-minor-free graphs for any fixed graph H, either via problem-specific techniques or by an appropriate generalization of the framework.

Background

The paper’s main distance-d result (Theorem 1.3) establishes, for K3,h-minor-free graphs, a randomized procedure that outputs an induced subgraph and a tree decomposition in which each bag intersects the distance-d neighborhood of any size-k pattern Z in only O(√k polylog k) vertices, enabling subexponential algorithms for problems involving distance constraints.

However, these guarantees are currently proved only for K3,h-minor-free graphs; the authors point out that extending such results to H-minor-free graphs for general H remains unresolved, and note that some problems may become intractable on broader classes. They explicitly pose the generalization question as open, suggesting either problem-specific methods or a broader framework extension.