Definition:Cut-Vertex

Definition

Let $G = \struct {V, E}$ be a connected graph.

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

Then $v$ is a cut-vertex of $G$ if and only if the vertex deletion $G - v$ is a vertex cut of $G$.

That is, such that $G - v$ is disconnected.

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

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

• Definition:Vertex Deletion
• Definition:Vertex Cut

Sources

• 1977: Gary Chartrand: Introductory Graph Theory ... (previous) ... (next): $\S 2.4$: Cut-Vertices and Bridges