Properties Ensuring Decidability of Homomorphism Indistinguishability

Characterize the properties of a graph class \mathcal{F} that ensure decidability of homomorphism indistinguishability over \mathcal{F}.

Background

The paper establishes decidability for every \mathsf{CMSO}_1-definable graph class of bounded cliquewidth, extending prior sufficient conditions for sparse graph classes. The conclusion asks for a broader structural characterization of graph classes for which homomorphism indistinguishability is decidable, in connection with a conjecture for minor-closed classes cited from earlier work.

References

Finally, our results are only the starting point for a theory of homomorphism indistinguishability over dense graph classes: Which properties of $\mathcal{F}$ ensure that $\equiv_{\mathcal{F}}$ is decidable, see the corresponding conjecture for minor-closed~$\mathcal{F}$?

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