Definition:Cut-Vertex

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.



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 $$DEF$$.