Polynomial-Time Complexity for Higher-Dimensional Dense Weisfeiler--Leman

Determine whether homomorphism indistinguishability over graphs of cliquewidth at most k admits a polynomial-time algorithm for any fixed k different from 2, or establish an appropriate complexity lower bound.

Background

The paper proves a conditional barrier to polynomial-time algorithms for homomorphism indistinguishability over cographs, which are exactly the graphs of cliquewidth at most 2. The authors note that their reduction does not establish analogous hardness for other fixed cliquewidth parameters. Thus, the complexity of the problem for fixed k different from 2 remains unresolved.