Characterization of the Collapse Function for Graph Classes

Determine for which graph classes \mathcal{F} there exists a function f\colon\mathbb{N}\to\mathbb{N} such that homomorphism indistinguishability over \mathcal{F} for $K_{t,t}$-subgraph-free graphs is equivalent to homomorphism indistinguishability over the members of \mathcal{F} that are $K_{f(t),f(t)}$-subgraph-free.

Background

The conclusion gives a precise formulation of the proposed dense-to-sparse collapse: for a graph class \mathcal{F}, one seeks a uniform function f controlling the biclique-exclusion threshold needed to recover homomorphism indistinguishability. The paper proves existence of such a function for the classes of graphs of bounded cliquewidth, but leaves the general characterization unresolved.

References

That is, for which graph classes $\mathcal{F}$, does there exist a function $f \colon \mathbb{N} \to \mathbb{N}$ such that $K_{t,t}$-subgraph-free graphs $G$ and $H$ are homomorphism indistinguishable over $\mathcal{F}$ if, and only if, they are homomorphism indistinguishable over the $K_{f(t),f(t)}$-subgraph-free graphs in $\mathcal{F}$?

— A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability  (2608.13382 - Curticapean et al., 13 Aug 2026) in Section 9 (Conclusion)