Generating cyclical dependency graphs from k-way partitions of DAGs representing boolean networks
Consider the following graph on three vertices:

$$u \to v \to w$$

This is acyclic.  If you set $A_1 = \{u,w\}$ and $A_2 = \{v\}$, you end up with a dependency graph that is cyclic: it has an edge $A_1 \to A_2$ (due to the edge $u \to v$) and an edge $A_2 \to A_1$ (due to the edge $v \to w$).