There are common definitions of [series-parallel (SP) graphs and digraphs](https://en.wikipedia.org/wiki/Series-parallel_graph): the basic idea is as follows. A SP graph (or digraph) has two distinguished vertices $s$ ("source") and $t$ ("target"). The graph with a single edge is SP by definition. Identifying the target of one SP graph with the source of another gives a "series" construction of a new SP graph. Identifying both sources and targets of two SP graphs gives a "parallel" construction of a new SP graph. 


For digraphs we might consider a third operation, in which the source of one graph is identified with the target of a second and vice versa ("feedback"), with (say) the first graph's source and target becoming those of the feedback graph. Is there a reference in which such digraphs are discussed?


Note that [since $K\_4$ (undirected) is not SP](http://dx.doi.org/10.1016/0022-247X(65)90125-3), the above-mentioned class of digraphs must be nontrivial.



Generalizing series-parallel digraphs with feedback
A slightly more general class is discussed in


Dan Dougherty, Claudio Gutiérrez.
Normal Forms and Reduction for Theories of Binary Relations.
LNCS 1833, pp 95-109 (2000)
(<http://link.springer.com/chapter/10.1007/10721975_7>)


In Definition 4 they introduce four operations:


1. parallel composition
2. series composition
3. converse: interchange source and target (without changing edge directions)
4. branching: set the target to be equal to the source


