How many possible graphs from 3 directed branch? I have three directed branches and I want to make a graph from these branches. 
Is it possible to calculate how many possible graphs can be created with the condition that each branch is only used once? The graph can be connected or disconnected. The maximum number of vertices is 6 for 3 given branches and you can't add more vertices.
Here are some example graphs:

 A: You're doing it right, you just want to be systematic, so you can be sure you've accounted for all the cases.  As you say in a comment (rephrased somewhat), if we consider the underlying undirected graph, there are either $1$, $2$ or $3$ connected components.
There is only once digraph with $3$ components, since we just have $3$ isolated edges, and it doesn't matter how they're oriented.
With two components, we have one isolated edge, and a path of length $2$.  It may help to remember that the sum of the out-degrees must equal the sum of the in-degrees.  Since each edge contributes $1$ to the indegrees and $1$ to the out-degrees, both these sums must be $3$.  Now the isolated edge has one vertex with in-degree $0$ and out-degree $1$ and one vertex with in-degree $1$ and out-degree $0$, so the sums for the path of length $2$ must both be $2$.  We have three possibilities:

*

*One vertex has in-degree $2$ and out-degree $0$, and the others each have in-degree $0$ and out-degree $1$

*One vertex has out-degree $2$ and in-degree $0$, and the others each have out-degree $0$ and in-degree $1$

*One vertex has in-degree $0$ and out-degree $1$, one has in-degree $1$ and out-degree $0$, and the third has in-degree $1$ and out-degree $1$.

It's easy to see that all these possibilities can occur.
When there is only one connected component, there are more possibilities.  The edges could form a path of length $3$, or they could be arranged in a triangle, or they could all meet at one vertex (like a "Y" shape.)  Now you have to figure out all the possible orientations for each.  Think about the in-degrees and out-degrees to see if you've covered all the possibilities.
