Some digraph classes that meet the directed path partition conjecture

Let $\lambda \left(D\right)$ denote the order of a longest path in a digraph $D$. For disjoint $A,B\subseteq V\left(D\right)$, if $A\cup B=V\left(D\right)$, we say that $(A,B)$ is a partition of $D$. The Directed Path Partition Conjecture (DPPC) states that for every digraph $D$ and every integer $q$ with $q<\lambda \left(D\right)$, there is a partition $(A,B)$ of $D$ such that $\lambda \left(D\left[A\right]\right)\le q$ and $\lambda \left(D\left[B\right]\right)\le \lambda \left(D\right)-q$.

Arroyo and Galeana-Sánchez (2013) proved that every strong 3-quasi-transitive digraph satisfies the DPPC. They also showed that the DPPC holds for compositions over an acyclic digraph with digraphs that meet the DPPC. In the paper, we show that non-strong 3-quasi-transitive digraphs and strong 4-transitive digraphs also satisfy the DPPC. Additionally, we show that the DPPC holds for the compositions over a unicyclic digraph with digraphs that satisfy the DPPC and certain other conditions. Furthermore, by applying different arguments, we show that the compositions over a cycle with arbitrary digraphs meets the DPPC.