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KVertexConnectedComponents[g,k]

gives the k-vertex-connected components of the graph g.

KVertexConnectedComponents[g,k,{v1,v2,…}]

gives the k-vertex-connected components that include at least one of the vertices v1, v2, … .

Details
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Basic Examples  
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Applications  
Properties & Relations  
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KVertexConnectedComponents

KVertexConnectedComponents[g,k]

gives the k-vertex-connected components of the graph g.

KVertexConnectedComponents[g,k,{v1,v2,…}]

gives the k-vertex-connected components that include at least one of the vertices v1, v2, … .

Details

  • KVertexConnectedComponents is also known as k-connected components.
  • KVertexConnectedComponents returns a list of components {c1,c2,…}, where each component ci is given as a list of vertices.
  • The component ci generates a maximal k-vertex-connected subgraph of g.
  • For an undirected graph, the vertices u and v are in the same component if there are at least k vertex-disjoint paths from u to v.
  • For a directed graph, the vertices u and v are in the same component if there are at least k vertex-disjoint directed paths from u to v and from v to u.

Examples

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Basic Examples  (2)

Find 2-connected components of a graph:

Wolfram Language code: g = [image];
Wolfram Language code: KVertexConnectedComponents[g, 2]

Show the 2-connected components:

Wolfram Language code: HighlightGraph[g, Subgraph[g, #]& /@ %]

Find the 2-connected components in a social network:

Wolfram Language code: g = ExampleData[{"NetworkGraph", "ZacharyKarateClub"}];
Wolfram Language code: KVertexConnectedComponents[g, 2]

Scope  (4)

KVertexConnectedComponents works with undirected graphs:

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

Select 2-connected components that include at least one of the specified vertices:

Wolfram Language code: KVertexConnectedComponents[[image], 2, {1, 2}]

Use patterns to select components:

Wolfram Language code: KVertexConnectedComponents[[image], 2, _ ? OddQ]

Works with large graphs:

Wolfram Language code: g = GridGraph[{10, 10, 10, 10}];
Wolfram Language code: Timing[Length[KVertexConnectedComponents[g, 2]]]
Wolfram Language code: VertexCount[g]

Applications  (1)

Highlight the k-connected components of a graph:

Wolfram Language code: g = [image];
Wolfram Language code: Table[HighlightGraph[g, Subgraph[g, #]& /@ KVertexConnectedComponents[g, k], PlotLabel -> "k=" <> ToString[k]], {k, 1, 2}]

Properties & Relations  (1)

A k-vertex-connected graph has a k-vertex-connected component as itself:

Wolfram Language code: g = [image];
Wolfram Language code: KVertexConnectedGraphQ[g, 2]
Wolfram Language code: KVertexConnectedComponents[g, 2]
Wolfram Research (2014), KVertexConnectedComponents, Wolfram Language function, https://reference.wolfram.com/language/ref/KVertexConnectedComponents.html.

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_kvertexconnectedcomponents, author="Wolfram Research", title="{KVertexConnectedComponents}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/KVertexConnectedComponents.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_kvertexconnectedcomponents, organization={Wolfram Research}, title={KVertexConnectedComponents}, year={2014}, url={https://reference.wolfram.com/language/ref/KVertexConnectedComponents.html}, note=[Accessed: 13-June-2026]}