Given an acyclic directed graph $G$, the set $E(G)$ of edges of $G$ equipped with the reachable order $\to$ is called the **edge poset** of $G$, where for two edges $e_1\to e_2$ means that there is a directed path starting from $e_1$ ending with $e_2$.

My question is that how can we characterize those edge posets of acyclic directed graphs, or under exactly what conditions a poset can be represented as the edge poset of an acyclic directed graph?

A closely related result is stated in the paper "Planarity and Edge Poset Dimension", at page 733, where the authors said that:

*Posets which are edge partial orders of graphs have been fully characterized (see , for example , [13]) . They are called N-free partial orders as they do not contain any N-shaped configuration . They are also characterized by the property that any maximal chain intersects any maximal antichain . We have the following property :

**PROPERTY 3** . Each N - free partial order is the edge - partial order of an unique
e - bipolarly oriented graph.*

This result means that the edge poset of an e-bipolarly oriented graph (or equivalently, an st-graph /PERT-graph) can be characterized by the property of N-freeness.

But I do not agree with the statement that posets which are edge partial orders of graphs have been fully characterized, which is what my question concerns.