Roberson’s homomorphism-distinguishing-closure conjecture

Prove that every graph class closed under taking minors and disjoint unions is homomorphism distinguishing closed, meaning that every graph outside the class can be separated by homomorphism counts from two graphs that are homomorphism indistinguishable over the class.

Background

The paper studies homomorphism indistinguishability over graph classes associated with bounded treewidth, bounded treedepth, and simultaneous width–depth decompositions. A graph class is homomorphism distinguishing closed when homomorphism indistinguishability over that class determines exactly the homomorphism-count tests supplied by graphs in the class.

The conjecture would imply that distinct minor- and disjoint-union-closed graph classes induce distinct homomorphism indistinguishability relations whenever one is properly contained in the other. The paper verifies the conjecture for the bounded-treedepth classes $\mathcal{TD}_q$ and the simultaneous width–depth classes studied there, but states that the general conjecture remains unresolved.