New sufficient conditions for the existence of kernels in digraphs

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. If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D, a semikernel modulo F, S of D is an independent set of vertices of D such that for every z∈V(D)−S for which there exists an Sz−arc of D−F, there also exists an zS−arc in D. In this work new sufficient conditions for a digraph to be a critical kernel imperfect digraph, in terms of semikernel modulo F, are presented.