We have the following definition of a Gammoid from Schrijver's "Combinatorial Optimization, Polyhedra and Efficiency" volume B:
"Take a directed graph $D = (V,A)$ and subsets $U$ and $S$ of $V$. For $X,Y \subseteq V$, call $X$ linked to $Y$ if $\lvert X \rvert = \lvert Y \rvert$ and $D$ has $\lvert X \rvert $ vertex-disjoint $X-Y$ paths. Let $\mathcal{I}$ be the collection of subsets $I$ of $S$ such that some subset of $U$ is linked to $I$. Then $M=(S,\mathcal{I})$ is a matroid, ... called a gammoid"
It's not too hard to come up with examples. Given an integer $k$, and subsets $X,Y \subseteq E$ let the function $g_j(k, X, Y)$ return any subset $Z \subseteq X\cap Y$ with at most $k$ elements. We can represent this as a digraph with a node for each element in $X \cup Y$. Each of these nodes have directed edges to an additional $k$ nodes. The function returns a set of up to $k$ vertex disjoint paths (each with a single edge) that represent the items selected from $X \cup Y$. The image of this function would represent the independent sets of a gammoid induced by this digraph and the $k$ nodes.
We can make a more complicated gammoid by cascading several of these graphs, and taking the set $U$ to be the set of sink nodes with no outgoing edges, while the nodes with no incident edges correspond to items in the ground set $S$. A simple example is given in the illustration below, but obviously the graph could be much more complicated.
My question is whether this is a complete description of a gammoid, and if not, whether someone can give an example of $(D,U)$ which induce a gammoid that cannot be represented this way.