A subset N of V(D) is said to be a kernel if it satisfies the following two properties: (1) for any two different vertices x and y in N there is no arc between them, and (2) for each vertex u in V(D)\(\setminus N\) there exists v in N such that (u,v) \(\in\) A(D). If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. In Galeana-Sánchez and Rojas-Monroy (Discrete Math, 275: 129–136, 2004) and Galeana-Sánchez and Rojas-Monroy (Discrete Math. 306: 1969–1974, 2006) the authors establish sufficient conditions to guarantee the kernel perfectness in digraphs, possibly infinite, where their set of arcs can be partitioned into at most two pre-transitive (resp. quasi-transitive) digraphs. In the present paper we consider those, also possibly infinite, digraphs where the set of arcs can be partitioned into at least three quasi-transitive (resp. pre-transitive) digraphs, and establish sufficient conditions to guarantee the kernel perfectness. In both cases we derive Richardson’s theorem, which states that every finite digraph without cycles of odd length has a kernel.

如果满足以下两个属性，则将V（D）的子集N称为内核：（1）对于N中的任意两个不同顶点x和y，它们之间都没有弧，并且（2）对于每个顶点u在V（D）\（\ setminus N \）中，在N中存在v使得（u，v）\（\ in \） A（D）。如果D的每个归纳子图都有一个核，则D据说是内核完美的有向图。在Galeana-Sánchez和Rojas-Monroy（Discrete Math，275：129–136，2004）和Galeana-Sánchez和Rojas-Monroy（Discrete Math。306：1969-1974，2006）中，作者建立了充分的条件来保证核完美。在有向图（可能是无限的）中，可以将它们的弧线集合最多划分为两个预传递（或准传递）图。在本文中，我们考虑了那些（也可能是无限的）有向图，其中弧集合可以划分为至少三个准传递（分别为过渡）图，并建立了充分的条件来保证核的完美。在这两种情况下，我们都推导出理查森定理，该理定理指出，每个无奇数周期的有限有向图都有一个核。