I am interested in how a structure of the following representation would be called or if there even is an established definition of such a thing.


The structure is similar to a graph. The representation shown here has the following meaning:


* There are nodes `A`, `B` and `C`.
* There are edges `x` and `y`.
* Node `A` has an edge `x` to node `B`.
* Node `C` has an edge `y` to edge `x`.


The first three sentences are typical descriptions for a graph. The last one does not fit into the definition of a graph. Here is a representation:



```
(A)---x------>(B)
          ^
          |
          |
          y
          |
          |
         (C)

```

This example could too be modeled using multiple graphs, lists or other structures, but I hope to find something more 'native'.



Is a graph with edges between edges and nodes still a graph?
I'm not aware of a name for such an object, but it can easily be modeled by a single vertex-colored directed graph $\Gamma$:


As vertex set of $\Gamma$, take the union of the vertices and edges of your original directed graph. Color the original vertices blue and the original edges red. Each original edge has a "source" and a "target", which in your case can be a vertex or an edge.


For the edge set of $\Gamma$, add two directed edges for each original edge: one from the source to the edge, and one from the edge to the target.


Now $\Gamma$ is a directed graph. (If your initial "thing" was an honest graph, then $\Gamma$ is bipartite.)


