Term for the graph of the cycles I am looking for the term which describes the graph G* of a graph G, composed by the same edges, but each cycle becomes the nodes of G*. 

How to call G*? (I grew up calling it "Star Graph" but now reality has changed, and star graph is something completely different https://en.wikipedia.org/wiki/Star_(graph_theory) )
https://en.wikipedia.org/wiki/Dual_graph   => "In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge."
https://en.wikipedia.org/wiki/Dual_graph   => "In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge."