Definition:Cut-Vertex

Definition

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

Let $v$ be a vertex of $G$.

Then $v$ is a cut-vertex of $G$ iff the vertex deletion $G - \left\{{v}\right\}$ is a vertex cut of $G$.

That is, such that $G - \left\{{v}\right\}$ is disconnected.

Thus, a cut-vertex of $G$ is a vertex of $G$ whose singleton is a vertex cut.

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 $DEF$.

Also see

• Vertex Deletion
• Vertex Cut

Sources

• Gary Chartrand: Introductory Graph Theory (1977): $\S 2.4$