Polynomial-time recognition of odd-cycle chord hypotheses

Determine whether the structural conditions in the Galeana-Sánchez–Neumann-Lara theorem and Theorem 2 can be checked in polynomial time.

Background

Checking whether every odd directed cycle satisfies a prescribed chord condition is itself algorithmically nontrivial. In addition to the unresolved question of computing kernels, the authors state that they do not know whether the hypotheses of the two kernel-existence theorems can be recognized in polynomial time.