Complexity of Homomorphism Indistinguishability over Bicliques

Determine whether homomorphism indistinguishability over the class of all bicliques is \mathsf{C}_=\mathsf{P}-complete.

Background

The paper proves conditional \mathsf{W[1]}-hardness against polynomial-time algorithms for homomorphism indistinguishability over all bicliques. It then observes that the authors expect the sharper classification \mathsf{C}_=\mathsf{P}-completeness, matching the known complexity for homomorphism indistinguishability over cliques, but do not establish it.

References

Moreover, we suspect that homomorphism indistinguishability over bicliques is $\mathsf{C}_=\mathsf{P}$-complete as is the case for cliques . Both problems remain open.

— A Dense Weisfeiler-Leman Algorithm for Deciding Bounded-Cliquewidth Homomorphism Indistinguishability  (2608.13382 - Curticapean et al., 13 Aug 2026) in Paragraph immediately preceding Theorem 8.1, Section 8 (Hardness of homomorphism indistinguishability over cographs)