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Bicomponents[g]

gives the biconnected components of the undirected graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
See Also
Tech Notes
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GraphUtilities`
GraphUtilities`

Bicomponents

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

Bicomponents[g]

gives the biconnected components of the undirected graph g.

Details and Options

  • Bicomponents functionality is now available in the built-in Wolfram Language function KVertexConnectedComponents.
  • To use Bicomponents, you first need to load the Graph Utilities Package using Needs["GraphUtilities`"].
  • A biconnected component is a maximal subgraph that has no cutpoint, where a cutpoint is a vertex v such that the subgraph becomes disconnected if v and all its edges are removed.
  • Bicomponents treats the input g as an undirected graph.

Examples

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Basic Examples  (2)

Wolfram Language code: Needs["GraphUtilities`"]

This shows that a simple line with two vertices is biconnected:

Wolfram Language code: g = {1 -> 2};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]
Wolfram Language code: Bicomponents[g]

Bicomponents has been superseded by KVertexConnectedComponents:

Wolfram Language code: g = Graph[{12}]
Wolfram Language code: KVertexConnectedComponents[g, 2]

Scope  (1)

Wolfram Language code: Needs["GraphUtilities`"]

This defines a small graph:

Wolfram Language code: g = {1 -> 2, 2 -> 3, 3 -> 4, 4 -> 5, 5 -> 6, 6 -> 15, 15 -> 7, 6 -> 1, 7 -> 8, 8 -> 9, 9 -> 10, 10 -> 11, 11 -> 12, 12 -> 7};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]

The graph has four bicomponents, one for each cycle and two for the line joining the cycles:

Wolfram Language code: Bicomponents[g]

Properties & Relations  (2)

Wolfram Language code: Needs["GraphUtilities`"]

This shows that a simple line with two vertices is biconnected:

Wolfram Language code: g = {1 -> 2};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]
Wolfram Language code: Bicomponents[g]

This defines a graph:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: g = {1 -> 2, 2 -> 3, 3 -> 4, 4 -> 1, 5 -> 6, 6 -> 7, 7 -> 5};
Wolfram Language code: GraphPlot[g, VertexLabeling -> True]

This shows its biconnected components and connected components:

Wolfram Language code: Bicomponents[g]

The result from WeakComponents is always smaller than that of Bicomponents:

Wolfram Language code: WeakComponents[g]
Wolfram Research (2007), Bicomponents, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/Bicomponents.html.

Text

Wolfram Research (2007), Bicomponents, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/Bicomponents.html.

CMS

Wolfram Language. 2007. "Bicomponents." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/GraphUtilities/ref/Bicomponents.html.

APA

Wolfram Language. (2007). Bicomponents. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/GraphUtilities/ref/Bicomponents.html

BibTeX

@misc{reference.wolfram_2026_bicomponents, author="Wolfram Research", title="{Bicomponents}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/Bicomponents.html}", note=[Accessed: 15-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_bicomponents, organization={Wolfram Research}, title={Bicomponents}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/Bicomponents.html}, note=[Accessed: 15-June-2026]}