Polynomial-time kernel computation for odd-cycle chord conditions

Determine whether a kernel can be computed in polynomial time for digraphs satisfying the hypotheses of either the Galeana-Sánchez–Neumann-Lara theorem or Theorem 2.

Background

Both the classical Galeana-Sánchez–Neumann-Lara theorem and Theorem 2 establish kernel existence from structural restrictions on odd directed cycles, but neither theorem is accompanied by an algorithmic result. The authors explicitly identify polynomial-time kernel computation under either set of hypotheses as unresolved.

References

For instance, we do not know whether a kernel can be computed in polynomial time under the conditions of either of these theorems.

— Revisiting classical results on kernels in digraphs  (2502.02482 - Langlois et al., 4 Feb 2025) in Section 3, subsection “Discussion,” second paragraph