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.