Directed Graphs on at most 9 vertices with some properties

Question

Let $\mathcal{F}$ be the family of strongly connected digraphs with $\leq 9$ vertices, up to isomorphism.

Asuume that a directed edge from vertex $v$ to $w$, if exists, is unique; and in this case, there is no directed edge from $w$ to $v$.

Is this family easy to enumerate, say using GAP or SAGE?

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Note: The elements of $\mathcal{F}$ have following properties:

(1) Each graph $G$ in $\mathcal{F}$ is connected and has $\leq 9$ vertices.

(2) For any distinct vertices $v,w$ of $G$, there is a directed path from $v$ to $w$.

(3) [Edit-from Smylic's comment]Every vertex has at least one incoming and one outgoing edge.

(4) If there is directed edge from $v$ to $w$, then no directed edge from $w$ to $v$.

Is it easy to obtain enumeration of the family $\mathcal{F}$, say using some softwares such as GAP, SAGE etc.?

I have handeled GAP for groups, but no idea whether we can handle directed graphs there.

Any suggestions for this problems? Also I would appreciate for suggestions of references on such graphs, their properties.

Answer 1

It took less that second to compute those up to 5 vertices, less than minute for up to 6 vertices. I guess you could run this up to 8 vertices in a week, but ninth vertex needs something more clever.

N = 5
def f(g):
    v = g.order()-1
    if v < 1: return True
    for v_ in g.neighbor_out_iterator(v):
        if g.has_edge(v_, v):
            return False
    return True
sum(1 for g in digraphs(N, augment='vertices', property=f) if g.is_strongly_connected())