Unit-multiplicative quasi-isometries for subdivision- and contraction-closed graph classes

Prove that for every class C of connected graphs closed under edge contraction and edge subdivision, and for all L,C, there exists C' such that every graph admitting an (L,C)-quasi-isometry to a member of C admits a (1,C')-quasi-isometry to a member of C.

Background

The paper asks when an arbitrary multiplicative quasi-isometry to a graph class can be improved to one with multiplicative constant 1 while retaining the target class. Closure under edge subdivision and contraction is identified as a necessary type of structural condition, motivating Conjecture 1.4. The conjecture is presented as a general principle, with the paper proving only special cases such as bounded path-width.

References

1.4. Conjecture: Let C be a class of connected graphs, closed under contracting edges and subdividing edges. For all L, C there exists C' such that if there is an (L, C)-quasi-isometry from a graph G to a member of C, then there is a (1, C')-quasi-isometry from G to a member of C.

— Coarse tree-width  (2501.09839 - Nguyen et al., 16 Jan 2025) in Conjecture 1.4, Section 1