Common generalization of two odd-cycle kernel theorems

Construct a common generalization of Duchet’s theorem, which assumes two reversible arcs on every odd directed cycle, and Theorem 2, which assumes one of three specified chord configurations on every odd directed cycle.

Background

The paper proves a new kernel-existence theorem extending the Galeana-Sánchez–Neumann-Lara result through several chord configurations. It then compares this theorem with Duchet’s independent result based on reversible arcs and explicitly leaves open whether both results can be subsumed by a single broader theorem.

References

We do not know whether there is a common generalization of this latter theorem and Theorem~\ref{thm:cordes_nous}.

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