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.