Efficient kernel computation under Proposition 3 conditions

Determine whether a kernel can be found in polynomial time in a red–blue digraph satisfying the two conditions of Proposition 3.

Background

Proposition 3 proves kernel existence for red–blue digraphs with no monochromatic directed cycle and an additional four-vertex arc condition. Its proof constructs a sequence of independent sets, but the authors cannot bound the number of sets considered polynomially, leaving the algorithmic complexity of finding a kernel unresolved.

References

We could not find any polynomial bound on the number of independent sets $I$ considered in the previous proof. As a result, we do not know whether any complexity result for finding a kernel in graphs satisfying conditions of Proposition~\ref{prop:4br} can be derived from the proof.

— Revisiting classical results on kernels in digraphs  (2502.02482 - Langlois et al., 4 Feb 2025) in Section 2, subsection “Discussion,” immediately after the proof of Proposition 3