<p>Well, <a href="http://www.graphclasses.org/classes/gc_39.html" rel="nofollow">split graphs</a> 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.</p>

CC BY-SA 3.0
2011-07-12T16:32:04.887Z
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
CC BY-SA 3.0
2011-07-12T16:27:11.227Z
2011-07-12T16:27:11.227Z
Anthony Labarre
<p>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? </p>

<p>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.</p>

CC BY-SA 3.0
2011-07-12T16:33:03.997Z
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
CC BY-SA 3.0
2011-07-12T17:04:35.687Z
Add that restriction into the question.
Dave Clarke
CC BY-SA 3.0
2011-07-12T20:03:49.017Z
So in other words the vertex set can be partitioned into a stable set and a set of cut-vertices?
Andrew D. King
CC BY-SA 3.0
2011-07-13T05:26:38.523Z
@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
<reference-request><graph-theory><co.combinatorics><terminology>
Standard/Formal name for the graph
singhsumit
# Standard/Formal name for the graph

(Asked by: username_0 on 2011-07-12)

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.

**Tags:** reference-request, graph-theory, co.combinatorics, terminology

> **Comments:**
> **username_0**: 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 $V_2$.  
> **username_1**: Add that restriction into the question.  
> **username_2**: So in other words the vertex set can be partitioned into a stable set and a set of cut-vertices?  
> **username_0**: @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.  

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## 1 answers

### Answer by username_3 on 2011-07-12. Score: 2

Well, [split graphs](http://www.graphclasses.org/classes/gc_39.html) 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.

> **Comments:**
> **username_0**: 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 $V_2$.  

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