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KVertexConnectedGraphQ[g,k]

yields True if the graph g is k-vertex-connected and False otherwise.

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KVertexConnectedGraphQ

KVertexConnectedGraphQ[g,k]

yields True if the graph g is k-vertex-connected and False otherwise.

Details

  • KVertexConnectedGraphQ is also known as k-connected.
  • A graph is k-vertex-connected if there are at least k vertex-disjoint paths between every pair of vertices.

Examples

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Basic Examples  (1)

Test whether a graph is 2-connected:

Wolfram Language code: Graph[{12, 23, 31}]
Wolfram Language code: KVertexConnectedGraphQ[%, 2]

Scope  (5)

Test undirected graphs:

Wolfram Language code: KVertexConnectedGraphQ[[image], 2]

Directed graphs:

Wolfram Language code: KVertexConnectedGraphQ[[image], 2]

Multigraphs:

Wolfram Language code: KVertexConnectedGraphQ[[image], 2]

Mixed graphs:

Wolfram Language code: KVertexConnectedGraphQ[[image], 2]

KVertexConnectedGraphQ gives False for anything that is not a k-connected graph:

Wolfram Language code: KVertexConnectedGraphQ[x, 1]

Properties & Relations  (2)

The complete graph is -connected:

Wolfram Language code: KVertexConnectedGraphQ[CompleteGraph[3], 2]

An undirected tree is 1-connected:

Wolfram Language code: TreeGraph[{12, 13, 14}]
Wolfram Language code: {KVertexConnectedGraphQ[%, 1], KVertexConnectedGraphQ[%, 2]}
Wolfram Research (2014), KVertexConnectedGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/KVertexConnectedGraphQ.html.

Text

Wolfram Research (2014), KVertexConnectedGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/KVertexConnectedGraphQ.html.

CMS

Wolfram Language. 2014. "KVertexConnectedGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/KVertexConnectedGraphQ.html.

APA

Wolfram Language. (2014). KVertexConnectedGraphQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/KVertexConnectedGraphQ.html

BibTeX

@misc{reference.wolfram_2026_kvertexconnectedgraphq, author="Wolfram Research", title="{KVertexConnectedGraphQ}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/KVertexConnectedGraphQ.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_kvertexconnectedgraphq, organization={Wolfram Research}, title={KVertexConnectedGraphQ}, year={2014}, url={https://reference.wolfram.com/language/ref/KVertexConnectedGraphQ.html}, note=[Accessed: 12-June-2026]}