I think you should be able to encode the axioms of a directed, acyclic graph by introducing a strict partial order. Say E(a, b) represents there is an edge from a to b. We introduce a strict partial order $R(\cdot, \cdot)$ and axiomatize acyclicality:
\begin{align*} \forall a, b~E(a, b) \rightarrow R(a, b) \hspace{2cm} &\text{(A1)} \\ \forall a, b, c~R(a, b) \land R(b, c) \rightarrow R(a, c) \hspace{2cm} &\text{(A2)} \\ \forall a~\neg R(a, a) \hspace{2cm} &\text{(A3)} \end{align*}
However, it seems like you can prove via compactness argument that acyclic graphs are not axiomatizable (like they did here). Which of these is correct?