Series-parallel DAG characterization
The two-connected series-parallel DAGs are also called 2-terminal series-parallel graphs. The graph in the figure is also known as Wheatstone graph and is known as the example graph for demonstrating Braess' paradox in algorithmic game theory.
It was shown [1, Theorem 5.3] that a DAG is 2-terminal series-parallel if and only if it does not contain a subgraph that is homeomorphic to the Wheatstone graph.
1 Pietro Cenciarelli, Daniele Gorla, Ivano Salvo: Inefficiencies in network models: A graph-theoretic perspective. Information Processing Letters 131 (2018) 44–50.