Improved bounds for the two-scale treewidth partition parameter

Determine whether the minimum threshold $\ell(p,q)$ has asymptotically better bounds than the currently known $O^*((pq)^9)$ bound, where every graph of treewidth at least $\ell(p,q)$ must contain pairwise disjoint connected subgraphs each of treewidth at least $p$ whose simultaneous contraction produces a minor of treewidth at least $q$.

Background

The paper defines $\ell(p,q)$ as the least integer such that every graph of treewidth at least $\ell(p,q)$ contains pairwise disjoint connected subgraphs $G_1,\ldots,G_n$, each with treewidth at least $p$, and contracting these subgraphs yields a minor of treewidth at least $q$. Using the Grid Minor Theorem, the authors obtain the bound $\ell(p,q)\in O^*((pq)^9)$.

The parameter captures the simultaneous presence of many locally complex subgraphs and a globally complex contracted minor. The authors note that existing partitioning results establish the local treewidth condition but do not establish the contraction condition, motivating the search for sharper bounds.