Let $G=(V,E)$ be an undirected graph. We form a graph $H=(V',E')$ from $G$ such that 


* $V' = V \cup \{ w\_e \mid e \in E \}$, and
* $E' = \{ aw\_e, bw\_e \mid ab = e \in E \} \cup \{ w\_e w\_f \mid e,f \text{ are adjacent edges in }G \}$.


Informally, $H$ is built from $G$ by subdividing each edge, and by putting an edge between two newly created vertices iff the corresponding edges are adjacent in $G$.


The above construction feels quite natural. Does it have a name? It seems like some kind of a variation of the line graph.



Is this special line graph of a graph a known concept?
Your graph is a subgraph of the [total graph](http://mathworld.wolfram.com/TotalGraph.html) of $G$. Both graphs have the same vertex set, and each edge in your graph is an edge in the total graph, but yours is missing edges for all the vertex-vertex adjacencies in $G$. 


