  Let $D$ be a digraph and let $\lambda(D)$ denote the number of vertices in a
longest path of $D$. For a pair of vertex-disjoint induced subdigraphs $A$ and
$B$ of $D$, we say that $(A,B)$ is a partition of $D$ if $V(A)\cup V(B)=V(D).$
The Path Partition Conjecture (PPC) states that for every digraph, $D$, and
every integer $q$ with $1\leq q\leq\lambda(D)-1$, there exists a partition
$(A,B)$ of $D$ such that $\lambda(A)\leq q$ and $\lambda(B)\leq\lambda(D)-q.$
Let $T$ be a digraph with vertex set $\{u_1,\dots, u_t\}$ and for every $i\in
[t]$, let $H_i$ be a digraph with vertex set $\{u_{i,j_i}\colon\,
  j_i\in [n_i]\}$. The {\em composition} $Q=T[H_1,\dots , H_t]$ of $T$ and
$H_1,\ldots, H_t$ is a digraph with vertex set $\{u_{i,j_i}\colon\, i\in [t],
j_i\in [n_i]\}$ and arc set $$A(Q)=\cup^t_{i=1}A(H_i)\cup
\{u_{i,j_i}u_{p,q_p}\colon\, u_iu_p\in A(T), j_i\in [n_i], q_p\in [n_p]\}.$$ We
say that $Q$ is acyclic {(semicomplete, respectively)} if $T$ is acyclic
{(semicomplete, respectively)}. In this paper, we introduce a conjecture
stronger than PPC using a property first studied by Bang-Jensen, Nielsen and
Yeo (2006) and show that the stronger conjecture holds for wide families of
acyclic and semicomplete compositions.
