I'm working on Ádám's conjecture about arc reversal problem on digraphs with at least one cycle. The thing is that Reid proved in 1984 the following theorem:
Suppose that $T$ is a strongly connected tournament such that the reversal of any single arc results in a strongly connected tournament, but that the reversal of some pair of arcs results in a tournament that is not strongly connected. Then there exists an arc on $T$ such that the reversal of this arc results in a tournament with strictly fewer directed cycles than $T$.
In order to prove this, he said that for existing such a tournament there must be a non-empty subset $S$ of $V(T)$ such that the number of arcs directed from $S$ to $V(T)-S$ is exactly two. I'm stuck on this part. It is easy to see that there should be at least two arcs of this type, but I can't see why couldn't be three.
Any kind of help will be appreciated.