On a directed acyclic graph $G=(V,E)$ the Minimum Path Cover (MPC) is the minimum number of paths that can be constructed on the DAG such that all vertices are covered by at least one path.

I know that there are practical circumstances where the MPC of a DAG is small, for example in genomics. I am wondering what kinds of conditions on the DAG would imply that the DAG’s MPC is small. One condition I know of is $sum_{v in V} deg_{out}(v)-deg_{in}(v) leq k$, where $k$ is some constant. However, I was looking for other such conditions, that restrict the MPC to size of $logN$ or $O(1)$, or similar.