Definition:Vertex Cut

Definition
Let $G$ be a graph.

A vertex cut of $G$ is a set of vertices $W \subseteq V \left({G}\right)$ such that the vertex deletion $G \setminus W$ is disconnected.

Example

 * Vertex-Cut.png

Removing $\left\{{B, C, F}\right\}$ would also remove all the dotted edges and leave this graph with two components:
 * one containing the vertices $A$ and $E$

and
 * one containing the vertices $D$, $G$, $H$, and $I$.

Thus $\left\{{B, C, F}\right\}$ is a vertex cut.

Also see

 * Vertex Deletion


 * Cut-Vertex: a vertex whose singleton is a vertex cut.