I was reading about line graph:
The line graph $L(G)$ of a simple graph $G$ is defined as follows: There is exactly one vertex $v(e)$ in $L(G)$ for each edge $e$ in G. For any two edges $e$ and $e'$ in $G$, $L(G)$ has an edge between $v(e)$ and $v(e')$, if and only if $e$ and $e'$ are incident with the same vertex in $G$.
Then I came across statement saying:
The line graph of a planar graph is not necessarily planar.
I see this is the case by considering e.g., the star graph $S_5$. However, can we generalize the property of the graph whose line graph results in non-planar graph. Obviously this must be the subgraphs whose line graph translates to either of Kuratowaski's graph $K_5$ or $K_{3,3}$. As pointed out, $L(K_5)=S_6$ which is planar. But $L(K_{3,3})$ is not planar (or can it be drawn as planar too?) as shown in figure below:
So is it like if the planar graph has a subgraph isomorphic to $S_5$, then its line graph is non planar (as it will contain $K_5$). Or is there anything more to this?