A digraph D is said to satisfy the k-Meyniel’s condition if each odd
directed cycle of D has at least k diagonals.
The study of the k-Meyniel’s condition has been a source of many
interesting problems, questions and results in the development of Kernel Theory.
In this paper we present a method to construct a large variety of
kernel-perfect (resp. critical kernel-imperfect) digraphs which satisfy
the k-Meyniel’s condition