TITLE: Graph isomorphism problem for minimally strongly connected digraphs
QUESTION [5 upvotes]: A minimally strongly connected digraph (MSC) is strongly connected (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 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).

REPLY [2 votes]: 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.