Polynomial-time recognition of the odd-cycle chord conditions

Determine whether the conditions requiring every odd directed cycle to contain one of the specified chord configurations in the Galeana-Sánchez–Neumann-Lara theorem and in Theorem 2 can be checked in polynomial time.

Background

The Galeana-Sánchez–Neumann-Lara theorem and Theorem 2 guarantee kernel existence under structural conditions imposed on every odd directed cycle. The paper explicitly observes that no polynomial-time recognition procedure is known for either the classical condition or the generalized condition.

References

Note that the complexity status of checking that a graph satisfies the condition of the theorem is unclear. The same situation holds for our extension.

— Revisiting classical results on kernels in digraphs  (2502.02482 - Langlois et al., 4 Feb 2025) in Section 1, subsection “Chords in odd holes,” immediately after Theorem 2