Is there an algorightm generating all digraphs with $n$ edges up to isomorphism whose underlying graph is not a tree? For example, for $n=3$, there are only two such digraphs, representable as $\text{Digraph}({[1,2],[2,3],[3,1]})$ and $\text{Digraph}({[1,2],[2,3],[1,3]})$.



Algorithm generating digraphs
There are algorithms that will directly enumerate non-isomorphic graphs, without duplicates, e.g. `geng` available as a part of [nauty](http://users.cecs.anu.edu.au/~bdm/nauty/) and it also have procedures to generate non-isomorphic digraphs with a fixed underlying graph, 
one is `directg` (see chapter 15 of [nauty manual](http://users.cecs.anu.edu.au/~bdm/nauty/nug26.pdf))


Edit: `nauty` also have various convenient interfaces, e.g. from Python ([SageMath](http://sagemath.org) and [pynauty](https://web.cs.dal.ca/~peter/software/pynauty/html/))


