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.

• 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 $V_2$. – singhsumit Jul 12 '11 at 16:33
• Add that restriction into the question. – Dave Clarke Jul 12 '11 at 17:04
• So in other words the vertex set can be partitioned into a stable set and a set of cut-vertices? – Andrew D. King Jul 12 '11 at 20:03
• @Andrew.. yes... the set $V_1$ is independent set and every vertex of $V_2$ is a cut vertex. When forming largest independent set, some of the vertices in $V_2$ $may$ join the vertices of $V_1$ to form that. – singhsumit Jul 13 '11 at 5:26

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.
• 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 $V_2$. – singhsumit Jul 12 '11 at 16:32