Some results on the structure of kernel-perfect and critical kernel-imperfect digraphs

Abstract

A kernel N of a digraph D is an independent set of vertices of D such that for every w∈V(D)∖N there exists an arc from w to N. The digraph D is said to be a kernel-perfect digraph when every induced subdigraph of D has a kernel. Minimal non kernel-perfect digraphs are called critical kernel-imperfect digraphs. The broader sufficient condition for the existence of kernels in digraphs known so far is that states: (1) If D is a digraph such that every odd cycle has two consecutive poles, then D is kernel-perfect.In this paper is studied the structure of critical kernel-imperfect digraphs which belong to a very large special classes of digraphs and many structural properties are obtained. As a consequence (1) is widely generalized in this class of digraphs, where the condition of the poles is requested only for odd cycles whose edges alternate in a set of arcs. As consequence, some classic results of kernel-perfect and finite critical kernel-imperfect digraphs are generalized for these classes of digraphs.