How do you call a graph that the bipartite property is satisfied for the one set of nodes and not the other?
Say you have two set of nodes $A$ and $B$ where there can be edges between nodes of set $A$ but there can be no edges between nodes of set $B$.Of course there can be edges between nodes of different sets.Is there a name for this type of graph?
This is more a characteristic of $B$. A subset $B$ of the nodes of a graph that have no edges between each other is called an independent set.
To further elaborate:
Every independent subset fulfills the criterion of the question, and every $B$ that fulfills the criterion of the question is inversely an independent subset.
Note that there are actually no criteria on $A$ other that it's the remaining graph, after removing $B$.