Classification of invariants satisfying Hedetniemi-type equality

Classify the homomorphism-monotone graph invariants ρ for which ρ(G × H) = min{ρ(G), ρ(H)} holds for all graphs G and H.

Background

For every homomorphism-monotone graph invariant ρ, the product inequality ρ(G × H) ≤ min{ρ(G), ρ(H)} holds automatically. The paper surveys invariants for which equality is known or remains unresolved and poses the general classification problem.