From a graph G I want to construct a graph (lets call it) G# with the following properties:
- Each node and each edge of G is a node of G#
- For each e in G connecting the nodes n1, n2 there exist the edges connecting e,n1 and e,n2
- no other edges exist in G#
G# is then bipartite with the nodes and edges of G being it's disjoint sets´. It's like creating the line graph of G and stopping after the creation of the "new nodes".
Is there a name for this special kind of graph?