Kernels in digraphs with covering number at most 3 (Q1861206)

A digraph \(D\) is said to be kernel-perfect if every proper induced subdigraph of \(D\) possesses a kernel. In 1998, the author (with her coauthor Xuliang Li) published two sufficient conditions for a digraph \(D\) to be kernel-perfect, concerning digraphs with the covering number at most three, containing only symmetrical directed triangles. In this paper the class of \(M\)-oriented digraphs is considered and a common generalization of both previous conditions is presented. The main result is the following one: Let every directed triangle of a digraph \(D\) have at least two symmetrical arcs. If each directed cycle of length five in \(D\) has two diagonals or three symmetrical arcs, then \(D\) is kernel-perfect.