Linear treewidth–Hadwiger bound for hereditary graph classes

Let $\mathcal{G}$ be a hereditary graph class. It is $(\operatorname{tw},\operatorname{had})$-bounded if there is a function $f$ such that $\operatorname{tw}(G)\leqslant f(\operatorname{had}(G))$ for every $G\in\mathcal{G}$. It is linearly $(\operatorname{tw},\operatorname{had})$-bounded if there is a constant $c$ such that $\operatorname{tw}(G)\leqslant c\,\operatorname{had}(G)$ for every $G\in\mathcal{G}$. Linear hereditary-class conjecture. Every $(\operatorname{tw},\operatorname{had})$-bounded hereditary graph class is linearly $(\operatorname{tw},\operatorname{had})$-bounded. The paper notes that this holds for circle graphs and proves related linear bounds for low-rank perturbations of circle graphs; the general hereditary-class claim remains open.