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FindMinimumCut[g]

gives the minimum cut of the graph g.

FindMinimumCut[{vw,…}]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
EdgeWeight  
Properties & Relations  
See Also
Related Guides
History
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FindMinimumCut

FindMinimumCut[g]

gives the minimum cut of the graph g.

FindMinimumCut[{vw,…}]

uses rules vw to specify the graph g.

Details and Options

  • A minimum k-cut of a graph g is a partition of vertices of g into k disjoint subsets with the smallest number of edges between them.
  • FindMinimumCut returns a list of the form {cmin,{c1,c2,…}}, where cmin is the value of a minimum cut found, and {c1,c2,…} is a partition of the vertices for which it is found.
  • For weighted graphs, FindMinimumCut gives a partition {c1,c2,…} with the smallest sum of edge weights possible between the sets ci.
  • The following option can be given:
  • EdgeWeight Automaticedge weight for each edge

Examples

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Basic Examples  (1)

Find the minimum cut:

Wolfram Language code: g = [image];
Wolfram Language code: FindMinimumCut[g]

Highlight the cut:

Wolfram Language code: HighlightGraph[g, Map[Subgraph[g, #]&, Last[%]]]

Scope  (7)

FindMinimumCut works with undirected graphs:

Wolfram Language code: FindMinimumCut[[image]]

Directed graphs:

Wolfram Language code: FindMinimumCut[[image]]

Weighted graphs:

Wolfram Language code: FindMinimumCut[[image]]

Multigraphs:

Wolfram Language code: FindMinimumCut[[image]]

Mixed graphs:

Wolfram Language code: FindMinimumCut[[image]]

Use rules to specify the graph:

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

FindMinimumCut works with large graphs:

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

Options  (1)

EdgeWeight  (1)

By default, the edge weight of an edge is taken to be its EdgeWeight property if available, otherwise 1:

Wolfram Language code: FindMinimumCut[[image]]

Use EdgeWeight->weights to set the edge weight:

Wolfram Language code: FindMinimumCut[[image], EdgeWeight -> Range[5]]

Properties & Relations  (3)

Use FindGraphPartition to find a cut with approximately equal-sized parts:

Wolfram Language code: g = [image];
Wolfram Language code: FindGraphPartition[g]

The minimum cut:

Wolfram Language code: Last[FindMinimumCut[g]]

EdgeConnectivity is the same as the value of a minimum cut:

Wolfram Language code: g = [image];
Wolfram Language code: {First[FindMinimumCut[g]], EdgeConnectivity[g]}

Use FindEdgeCut to obtain edges between cut sets:

Wolfram Language code: g = [image];
Wolfram Language code: FindEdgeCut[g]

Highlight the edges and cut sets:

Wolfram Language code: HighlightGraph[g, Join[{%}, FindMinimumCut[g][[2]]]]
Wolfram Research (2012), FindMinimumCut, Wolfram Language function, https://reference.wolfram.com/language/ref/FindMinimumCut.html (updated 2015).

Text

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

CMS

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

APA

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

BibTeX

@misc{reference.wolfram_2026_findminimumcut, author="Wolfram Research", title="{FindMinimumCut}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/FindMinimumCut.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_findminimumcut, organization={Wolfram Research}, title={FindMinimumCut}, year={2015}, url={https://reference.wolfram.com/language/ref/FindMinimumCut.html}, note=[Accessed: 12-June-2026]}