Nguyen–Scott–Seymour coarse tree-width conjecture

Establish whether there exists a constant k such that every graph G that admits a quasi-isometry to a graph of tree-width at most two also admits a quasi-isometry with additive distortion to a graph of tree-width at most k.

Background

The paper studies quasi-isometries with purely additive distortion between graph classes and proves that every K_{2,3}-induced minor-free graph admits such a quasi-isometry to a graph of tree-width at most two. This establishes the stated conjecture for the subclass of K_{2,3}-induced minor-free graphs, but the conjecture itself concerns all graphs that are quasi-isometric to graphs of tree-width at most two and is not resolved in the paper at that level of generality.

References

There is a constant $k$, such that if a graph $G$ admits a quasi-isometry to a graph of tree-width at most two, then $G$ admits a quasi-isometry with additive distortion to a graph with tree-width at most $k$.

— $K_{2,3}$-induced minor-free graphs admit quasi-isometry with additive distortion to graphs of tree-width at most two  (2503.00798 - Chakraborty, 2 Mar 2025) in Conjecture 1, Introduction