DAG decomposition into "parallel" components Let's consider a DAG (directed acyclic graph) with single source and single sink, for which the set of all its paths from the source to the sink can be split into subsets, such that any two paths from different subsets don't have common arcs.
This DAG can be decomposed into a number of "parallel" components, which have only the source and the sink in common. Does this decomposition have a name?
 A: In the context of series-parallel graphs, we simply say that the graph is a parallel composition of those components.
Specifically, given digraphs $D_1, \dots, D_n$ each with a designated source $s_i$ and sink $t_i$, their parallel composition is obtained in the following way:

*

*Take the disjoint union of $D_1$ through $D_n$.

*Merge the $n$ vertices $s_1, \dots, s_n$ into a single source $s$ with an edge $s \to v$ whenever we had an edge $s_i \to v$ for any $i$.

*Merge the $n$ vertices $t_1, \dots, t_n$ into a single sink $t$ with an edge $v \to t$ whenever we had an edge $v \to t_i$ for any $i$.

Usually, our goal is to define series-parallel digraphs as graphs that can be obtained from many directed $K_2$'s by this operation and by series composition. But parallel composition can be defined even when the individual digraphs we're composing don't have this structure.
(If you look at the Wikipedia link, it gives a definition for undirected graphs, but mentions that it extends to series-parallel digraphs, which is what I've done here.)
