Given a graph $G$, its line graph, denoted $L(G)$, is the graph whose vertices are the edges of $G$ and where two edges of $G$ are adjacent in $L(G)$ if they are incident to each other, i.e., they share some endpoint. I am interested in the graph, let's call it $L'(G)$, whose vertices are the edges of $G$, two being adjacent if they are incident but not part of a triangle in $G$, i.e., the two vertices not shared by the edges are not adjacent in $G$. Obviously, for a triangle free graph $L(G)$ and $L'(G)$ are the same, but in general $L'(G)$ is just some spanning subgraph of $L(G)$.

Does anyone know if this construction has come up in the literature before?