Richardson's theorem in quasi-transitive and pre-transitive digraphs

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)\N A(D). If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. In Galeana-Sanchez and Rojas-Monroy (Discrete Math, 275: 129-136, 2004) and Galeana-Sanchez 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.