I want to generate all strongly connected tournament of size $n \in \{4, 11\}$.   

As a strongly connected tournament has an hamiltonian path I may assume that $v\_i v\_{i+1}$ is always an arc, and $v\_n v\_1$ is an arc. Thus, I have to decide the outcome of $m:= n\frac{(n-1)}{2} -n$ games, ie I have $2^m$ tournaments to study. \n
However, it is clear some tournaments are generated multiple times.  

From what I read here on similar questions it seems that finding an optimal way to do it is probably a very complicated question; but:


1. Do you have guesses on what additional assumptions I may take?
2. A completely different idea on how to do abord the question?


Bonus question: if I do not change my algorithm, can I know how many times each tournament is generated?



Generate all strongly connected tournament
Brendan McKays program “gentourng” can generate all tournaments up to isomorphism - I recently used it for all 11-vertex tournaments.


It is part of the nauty package:


[https://users.cecs.anu.edu.au/~bdm/nauty/](https://users.cecs.anu.edu.au/%7Ebdm/nauty/)


