Superlogarithmic-to-logarithmic treewidth transition

Determine for which families of graphs \(\mathcal H\) there exists a hereditary graph class \(\mathcal C\) of superlogarithmic treewidth such that every \(H\)-free graph in \(\mathcal C\), for \(H\in\mathcal H\), has treewidth at most logarithmic in its number of vertices.

Background

The paper proves that for every graph HH, one can construct a hereditary class of unbounded treewidth whose HH-free graphs have bounded treewidth. It then asks whether an analogous phenomenon holds at the finer scale separating superlogarithmic treewidth from logarithmic treewidth.

The displayed problem asks for a classification of the graph families that can enforce this weaker transition. The authors explicitly note that the problem remains unresolved even when the family consists of a single graph.

References

For which families $\mathcal H$ is there a hereditary class $\mathcal C$ of superlogarithmic treewidth such that, for every $H \in \mathcal H$, the $H$-free graphs of $\mathcal C$ have at most logarithmic treewidth? \Cref{p:logarithmic-tw} is already open for singleton families $\mathcal H = {H}$.

— Every Graph is Essential to Large Treewidth  (2502.14775 - Alecu et al., 20 Feb 2025) in Problem 1, Section ‘Treewidth logarithmically bounded in the number of vertices’