Definition:Cut-Vertex

Definition
Let $G = \left({V, E}\right)$ be a connected graph.

Let $v \in V$ be a vertex of $G$ such that $G - v$ is disconnected.

Then $v$ is known as a cut-vertex of $G$.

In this context, $G - v$ signifies the graph $G$ with the vertex $v$removed, along with all the edges incident to it.

Example
In the graph below, $C$ is a cut-vertex.


 * Cut-Vertex.png

The edges $AC, BC, CD, CF$ are the edges which would be removed if $C$ were cut.

The graph would be separated into the two components $AB$ and $DEF$.