Quasi-isometry reduction for graph classes closed under contractions and subdivisions

Prove that for every class of connected graphs closed under contracting edges and subdividing edges, and for all natural numbers L and C, there exists a natural number C' such that whenever a graph G is (L,C)-quasi-isometric to a graph in that class, G is (1,C')-quasi-isometric to a graph in the same class.

Background

The paper disproves the stronger conjecture that, for every pair of graphs related by an (L,C)-quasi-isometry, an edge weighting of the target graph can always eliminate the multiplicative distortion factor while changing only the additive constant. The authors then identify a weaker graph-class version that remains unresolved.

The proposed statement concerns any class of connected graphs stable under edge contraction and edge subdivision. It asks whether quasi-isometry to a member of such a class can always be replaced by a slack-isometry, with multiplicative factor 1, to another member of the same class and with an additive constant depending only on the original distortion parameters L and C. The authors single out connected planar graphs as a particularly interesting special case, previously conjectured by Georgakopoulos in private communication.

References

Finally, while \cref{MainConjecture} is false, we would like to point out the following weakening of the conjecture which remains open.

\begin{conj}[]\label{ConjectureGraphClasses} Let G be a class of connected graphs closed under contracting edges and subdividing edges. For all L,C\in N, there exists C'\in N such that if a graph G is (L, C)-quasi-isometric to a graph in G, then G is (1, C')-quasi-isometric to a graph in G. \end{conj}

— Quasi-isometries between graphs with variable edge lengths  (2503.07448 - Davies et al., 10 Mar 2025) in Section Conclusion, Conjecture GraphClasses (labelled Conjecture \ref{ConjectureGraphClasses})