# Graph vertex set with a certain property

Let $G$ be a graph and let $V$ be a set of vertices with the following property: If a vertex $v$ is connected to every $u\in V$, then $v$ has to be in $V$. Does such $V$ have a (standard) name? Note that $V$ may not be a maximal clique, since it may happen that the induced graph on $V$ is not complete.

• It seems to me like you can take any set of nodes $V_0$ and let $V_n$ be the union of $V_{n-1}$ with the set of nodes that are neighbour to all vertices in $V_{n-1}$. Eventually you get a set like yours. – Arthur Jun 22 '15 at 11:00
• @Arthur Doesn't $V_1$ allready have the stated property? Let $v \in V(G)$ be connected to all vertices in $V_1$. Then $v$ is connected to all vertices in $V_0$ (as $V_0 \subseteq V_1$), hence $v \in V_1$. – martini Jun 22 '15 at 11:43
• @martini You're right. – Arthur Jun 22 '15 at 11:46
• It's not clear, whether the set $V$ is well-defined... – DVD Jun 24 '15 at 0:15

You can use the definition of simplical vertices. A simplical vertex $v$ is a vertex whose neighbors $N(v)$ form a clique. Thus, the set of vertices $V =\{v\} \cup N(v)$ is the set that you are interested in.