A minimum vertex cut of a graph is a vertex cut of
smallest possible size.

A vertex cut set of size 1 in a connected graph
corresponds to an articulation vertex.

The size of a minimum vertex cut in a connected graph gives the vertex
connectivity .

Complete graphs have no vertex cuts since there is no subset of vertices whose removal disconnected a complete
graph.

A single minimum vertex cut of a connected graph can be found in the Wolfram
Language using the function `FindVertexCut`[*G*].

## See also

Articulation Vertex,

Disconnected Graph,

Edge Cut,

*k*-Connected
Graph,

Mincut,

Minimal
Vertex Cut,

Vertex Connectivity,

Vertex
Cut
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## References

Skiena, S. *Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica.* Reading,
MA: Addison-Wesley, 1990.West, D. B. *Introduction
to Graph Theory, 2nd ed.* Englewood Cliffs, NJ: Prentice-Hall, p. 149,
2000.
## Cite this as:

Weisstein, Eric W. "Minimum Vertex Cut."
From *MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/MinimumVertexCut.html

## Subject classifications