Name for the tree formed after collapsing cycles What do you call a structure which becomes a tree after collapsing cycles (so the new vertices are the old faces)?
For instance, the digraph (given by an NFA; ignore edge labels please) below becomes a tree
$\{a,b,c,d\}$ with edges $(a,b), (b,c), (b,d)$ via:
\begin{eqnarray*}
a\mapsto&amp;\text{the cycle }&amp;S_0,S_1,S_2,S_3,S_4,S_5,S_6,S_7,S_0
\\
b\mapsto &amp;\text{the cycle }&amp;S_5,S_6,S_7,S_8,S_9,S_{10},S_{11},S_{12},S_5\\
c\mapsto &amp;\text{the cycle }&amp;S_8,S_9,S_8\\
d\mapsto &amp;\text{the cycle }&amp;S_{10},S_{11},S_{12},S_{10}
\end{eqnarray*}
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)