Name for the tree formed after collapsing cycles
For any given undirected graph $G$, the cycle graph $C(G)$ has vertices which correspond to the chordless cycles of $G$, and two distinct vertices of $C(G)$ are adjacent if and only if the corresponding chordless cycles have at least one edge in common. [1] 

Judging from your example, you are interested in a similar notion for a directed graph. 

[1] S.V. Gervacio. Cycle graphs. Lect. Notes Math. 1073, 279-293 (1984)