Given a directed graph $C$ that only contains a directed cycle of length $L$ (and all resulting sub-cycles), that visits each node at least once,
where $V$ is a fixed set of vertices and $D$ is a set of directed edges, and:
$$|V| = N$$
$$|D| = L$$
$$ 1 \leq \deg^+(v) \leq 2, v \in V$$
How many directed graphs $G=(V,E)$, where $E$ is a set of directed edges and,
$$ \deg^+(v) = 2, v \in V$$
contain $C$? I am not considering $G$ that contains a graph isomorphic to $C$, I am only interested in $G$ that contain exactly $C$.
That is, if F is the set of all G that contain exactly C as a subgraph, and
$$ n = |F| $$
what is n?