I am wondering if there is a name for a graph transformation that involves splitting up vertices in a directed graph into multiple vertices (one for each incoming and one for each outgoing edge of the original vertex), and then adding edges between these split up vertices in order to create an edge for each possible 'turn' in the original graph while still maintaining the original edges of the graph.
A simple algorithm for applying this transformation to a graph $G(E,V)$ would go as follows:
- For each vertex $v$ in $G$ create a line graph using the edges that are connected to $v$
- For each edge $e$ in $G$ add an edge $e'$ between the two line graphs that were created using the vertices in the original graph that connect to $e$.
- $e'$ is placed between the two vertices in the new line graphs that were created from $e$
Below is a visual representation of this graph transformation done on a small graph: