Suppose that the graph $H$ is an induced sub-graph of the graph $G$.
Is it the case that the line graph of $H$, $L(H)$ is an induced sub-graph of the line graph of $G$, $L(G)$.
Conversely, when $L(H)$ is an induced sub-graph of $L(G)$, is $H$ an induced sub-graph of $G$.
I am confident that 1. always holds. I also think that, barring subtleties with the claw graph and $K_3$, 2. holds.
I have searched for a proof or even a mention of this but was unable to find one.
Firstly, is it true that $H$ is an induced sub-graph of $G$ if and only if $L(H)$ is an induced sub-graph of $L(G)$? Secondly, if so, is there a reference to a proof of this?