Let $G = (V, E)$ be a digraph. We define a "rho of length $k$" as a finite sequence of $k$ vertices, each a neighbor of its predecessor, where exactly one of the following two conditions hold

1. All $k$ vertices are distinct, and the last vertex has no neighbors.
2. The first $k-1$ vertices of the sequence are distinct, and the last vertex is one of the first $k-1$ members of the sequence.

Let $R(G)$ be the set of all rhos contained in $G$.
Given $\rho \in R(G)$ by abuse of notation define $E(\rho)$
to be the edges of $\rho$. Extending the notation $R$, let $R(G; v)$ be the set of all rhos in $G$ whose initial vertex is $v$. We describe the "Rho Cover Problem" for a graph $G$ and vertex $v$ as the problem of finding the smallest set of rhos which start at $v$ and cover the edges of $G$.

Mathematically speaking, we can exhaustively check all subsets of $\{E(\rho) : \rho \in R(G; v)\}$, note the ones which cover $E$, and return one of minimal cardinality. The conditions to show such a subset exists were addressed David Eppstein: for all edges $(w,z)$ in $G$ there must be a path from $v$ to $w$.

From a computer science perspective you can now ask if there is an "easy" solution. My expectation is that answering this question would involve either finding a polynomial time algorithm for this problem (in which case it would be "easy") or showing the problem is NP-complete (in which case it would be "hard"). This would be best answered in either [cs](http://cs.stackexchange.com/) or [cstheory](http://cstheory.stackexchange.com/) depending on the caliber of the question (which I cannot judge).

From a programming perspective, you seem to be asking for a practical way to solve this problem for graphs with approximately 10 vertices. With so few vertices you can enumerate every rho (via a depth first search) and consider the Rho Cover Problem as a special instance of the Set Cover Problem. As noted in the [Wikipedia article on the Set Cover Problem](https://en.wikipedia.org/wiki/Set_cover_problem) this can be expressed as an Integer Linear Programming Problem. To solve such a problem I would recommend an academic licence of [Gurobi](http://www.gurobi.com/). I have found it to be an *extremely* powerful piece of software designed first and foremost for optimization.

Best of luck.