Is this graph transformation G\_1 to G\_2 efficiently computable?


1. All vertices in G\_1 are unique edges in G\_2
2. Adjacent vertices in G\_1 are adjacent edges in G\_2


The inverse transformation (edges to vertices) is trivial.


I can encode it as CSP, but I am not sure solving is tractable:


adjacent V1,V2 in G\_1 to (V1',V1''), (V2',V2'') in G\_2, must share a vertex, constraints: V1'=V2' / V1'=V2'' / V1''=V2' / V1''=V2'' and some disequations.


References, search terms will be apreciated. Thank you.



Is the graph transformation vertices to edges efficiently computable?
This is only possible if $G\_1$ is the [line graph](http://en.wikipedia.org/wiki/Line_graph) of $G\_2$. Not all graphs are line graphs.



> 
> A graph $G$ is the line graph of some other graph, if and only if it is possible to find a collection of cliques in $G$, partitioning the edges of $G$, such that each vertex of $G$ belongs to at most two of the cliques.
> 
> 
> 


[Here](http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0F-45FCW4T-37&_user=10&_coverDate=10%2F31%2F1973&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=bde3a2a5cadbc74f8878ccdfc7ac7989&searchtype=a) you can find am algorithm to recover the original graph from it's line graph. 


