Counting extensions of the pattern-sparse decision framework

Develop counting analogues of the presented decision algorithms by extending the pattern-sparse tree-decomposition sampling approach to counting versions of the problems, achieving comparable guarantees (e.g., subexponential dependence on k) to those established for the decision setting.

Background

The paper focuses on decision problems, providing randomized constructions of induced subgraphs and tree decompositions that enable subexponential-time algorithms in H-minor-free graphs (and, for distance constraints, in K3,h-minor-free graphs).

The authors explicitly defer counting versions—central in prior work by Nederlof—to future work, identifying the development of counting counterparts to their decision framework as an open direction.