A minimally strongly connected digraph (MSC) is [strongly connected](https://en.wikipedia.org/wiki/Strongly_connected_component) (SC), while removal of any arc destroys this. That is, between any two vertices a, b there exists a directed path from a to b, while removal of any arc (a,b) renders b unreachable from a.


What is known about graph isomorphism for MSCs?


For example, is decision of graph isomorphism between MSCs GI-complete ("graph-isomorphism complete" -- as hard as deciding isomorphism between general digraphs), or is it poly-time decidable? 
There seems to be no literature on this subject.


Another question is about determining UMS(n), the number of isomorphism types of MSCs for n vertices. 
In the paper [Minimal strong digraphs](http://www.sciencedirect.com/science/article/pii/S0012365X11005073) it is mentioned that UMS(n), in other words, the number of unlabeled MSC digraphs with n vertices, is "unknown": does there exist at least a superpolynomial lower bound for UMS(n)? 
Again, there seems to be nothing in the literature on this subject (so any information is welcome).



Graph isomorphism problem for minimally strongly connected digraphs
Isomorphism of MSC digraphs is *isomorphism-complete*. Consider two connected undirected graphs $G,H$ with no vertices of degree 1. It is routine to see that connectivity and minimum degree at least 2 won't help you to determine whether $G$ and $H$ are isomorphic.


Now convert $G,H$ to digraphs $G',H'$ by replacing each edge with a directed 4-cycle using two new vertices. Precisely, replace each edge $\lbrace v,w\rbrace$ by the 4-cycle $v\to x\_{v,w}\to w\to y\_{v,w}\to v$, where $x\_{v,w},y\_{v,w}$ are new vertices not used anywhere else. Now note that $G',H'$ are MSC digraphs which are isomorphic iff $G,H$ are isomorphic.


On the second question I'm not aware of a complete answer. Some small numbers are on [OEIS](http://oeis.org/A130756). 


