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VertexConnectivity[g]

gives the vertex connectivity of the graph g.

VertexConnectivity[g,s,t]

gives the s-t vertex connectivity of the graph g.

VertexConnectivity[{vw,…},…]

uses rules vw to specify the graph g.

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
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VertexConnectivity

VertexConnectivity[g]

gives the vertex connectivity of the graph g.

VertexConnectivity[g,s,t]

gives the s-t vertex connectivity of the graph g.

VertexConnectivity[{vw,…},…]

uses rules vw to specify the graph g.

Details

  • VertexConnectivity is also known as connectivity or point connectivity.
  • The vertex connectivity of a graph g is the smallest number of vertices whose deletion from g either disconnects g or reduces it to a single vertex graph.
  • The s-t vertex connectivity is the smallest number of vertices whose deletion from g disconnects g with s and t in two different connected components.
  • For a disconnected graph, VertexConnectivity will return 0.
  • VertexConnectivity works with undirected graphs, directed graphs, multigraphs, and mixed graphs.

Examples

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Basic Examples  (2)

Find the vertex connectivity:

Wolfram Language code: VertexConnectivity[[image]]

Find the vertex connectivity between two vertices:

Wolfram Language code: VertexConnectivity[[image], 2, 5]

Scope  (6)

VertexConnectivity works on undirected graphs:

Wolfram Language code: VertexConnectivity[[image]]

Directed graphs:

Wolfram Language code: VertexConnectivity[[image]]

Multigraphs:

Wolfram Language code: VertexConnectivity[[image]]

Mixed graphs:

Wolfram Language code: VertexConnectivity[[image]]

Use rules to specify the graph:

Wolfram Language code: VertexConnectivity[{1 -> 2, 2 -> 3, 4 -> 3, 6 -> 1, 6 -> 5, 5 -> 2, 5 -> 4, 2 -> 6, 3 -> 5}]

VertexConnectivity works on large graphs:

Wolfram Language code: g = GridGraph[{10, 10, 10}];
Wolfram Language code: VertexConnectivity[g]//Timing

Properties & Relations  (2)

The connectivity of a complete graph is :

Wolfram Language code: VertexConnectivity[CompleteGraph[4]]

Use FindVertexCut to compute the vertex connectivity:

Wolfram Language code: g = [image];
Wolfram Language code: {Length[FindVertexCut[g]], VertexConnectivity[g]}
Wolfram Research (2012), VertexConnectivity, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexConnectivity.html (updated 2015).

Text

Wolfram Research (2012), VertexConnectivity, Wolfram Language function, https://reference.wolfram.com/language/ref/VertexConnectivity.html (updated 2015).

CMS

Wolfram Language. 2012. "VertexConnectivity." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/VertexConnectivity.html.

APA

Wolfram Language. (2012). VertexConnectivity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/VertexConnectivity.html

BibTeX

@misc{reference.wolfram_2026_vertexconnectivity, author="Wolfram Research", title="{VertexConnectivity}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/VertexConnectivity.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_vertexconnectivity, organization={Wolfram Research}, title={VertexConnectivity}, year={2015}, url={https://reference.wolfram.com/language/ref/VertexConnectivity.html}, note=[Accessed: 13-June-2026]}