One can represent every group as a directed graph with colored edges (its Cayley graph). Identifying the colors of the edges with specific vertices of the graph (its generators), one ends up with a simple, uncolored digraph. There are interesting questions concerning such digraphs.
I'd like to ask a similar question: Instead of investigating how every group can be represented by a digraph, I'd like to know:
How can digraphs be represented by undirected graphs?
When are such representations "unique"?
How can the undirected graphs be characterized that represent digraphs?
How can - eventually - the digraph be reconstructed from its representation as an undirected graph?
One approach is to represent every arrow of the digraph by an extra-vertex and giving it a "direction" - represented by another extra-vertex. Very soon, one stumbles over an example of an undirected graph, that represents two very different digraphs.
(The labels are only for facilitating the reading and comparison of the graphs, they are not essential and are to be ignored.)
But most surely this is only an exception due to the small number of vertices and arrows. I believe, that in general – at least for almost all connected digraphs – the representation will be unique.
Or how can the cases of non-uniqueness be characterized specifically?
Here is another example (to be compared with the first diagram above):