Is a directed graph uniquely determined by the in/out degree of each node? I never really thought of this problem. If we have two directed graphs $A$ and $B$ with the same set of nodes $V$, and we know that the in/out degree of each node is the same in the $A$ and $B$, is it possible for $A$ and $B$ to have a different set of edges? In other words, does the in/out degree of each node guarantee the uniqueness of a graph?
I have tried to find a counter-example of two graphs which demonstrate this property (each node has same in/out degree in $A$ and $B$, but the edge set of the two graphs is different), but have come up empty handed thus far. Perhaps next would be finding a proof, or a counter-example.
As a further edit, the graphs I have in mind are connected.

Consider the graph on A, B, C with directed edges AB, BC, CA, and the graph with AC, BA, CB. 
If you want the graphs nonisomorphic, take AB, BC, DC, ED, FC versus AC, DC, BD, EB, FC. 
