Cut-Vertex divides Graph into Two or More Components

Theorem

Let $G$ be a graph.

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

Then the vertex deletion $G - v$ contains $2$ or more components.

Proof

By definition of cut-vertex, $G - v$ contains at least $2$ components.

That it can contain more components than $2$ is best proved by illustration:

$\blacksquare$

Sources

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