Standard/Formal name for the graph Given a connected graph $G =(V_1,V_2,E)$, such that there are no edges among the vertices in set $V_1$, however the other set $V_2$ can have edges in itself. There is actually a restriction for $V_2$, which is that we cannot remove any vertex in $V_2$ such that graph $G$ is still connected. Therefore a minimality constraint is there for $V_2$. Is there a standard name for this graph? 

The need for it is because im writing a manuscript which includes a result on such graphs and i need to refer this graph with some name. So, i was wondering if there was some name for it. Obviously i couldn't find it. Any other suggestion/comment is also appreciated.
Well, split graphs correspond to the case where $V_1$ is an independent set and $V_2$ forms a clique, but I'm not aware of another name in the case where $V_2$ has no particular restriction.