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TopologicalSort[g]

gives a list of vertices of g in topologically sorted order for a directed acyclic graph g.

TopologicalSort[{vw,…}]

uses rules vw to specify the graph g.

Details
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Examples  
Basic Examples  
Scope  
Applications  
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TopologicalSort

TopologicalSort[g]

gives a list of vertices of g in topologically sorted order for a directed acyclic graph g.

TopologicalSort[{vw,…}]

uses rules vw to specify the graph g.

Details

  • TopologicalSort is also known as topological ranking or topological ordering.
  • A list of vertices is topologically sorted if u precedes v for each edge uv.

Examples

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Basic Examples  (1)

Find the topological order of vertices:

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

Scope  (6)

TopologicalSort works with directed graphs:

Wolfram Language code: TopologicalSort[[image]]

Weighted graphs:

Wolfram Language code: TopologicalSort[[image]]

Multigraphs:

Wolfram Language code: TopologicalSort[[image]]

TopologicalSort only works with acyclic graphs:

Wolfram Language code: TopologicalSort /@ {[image], [image]}

Use rules to specify the graph:

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

Works with large graphs:

Wolfram Language code: adj = SparseArray[{i_, j_} /; j > i :> If[RandomReal[] < 0.05, 1, 0], {10^3, 10^3}]; ind = RandomSample[Range[Length[adj]], Length[adj]]; g = AdjacencyGraph[adj[[ind, ind]]];
Wolfram Language code: Timing[TopologicalSort[g]//Length]
Wolfram Language code: {VertexCount[g], EdgeCount[g]}

Applications  (1)

Sort strongly connected components instead of vertices when there are cycles:

Wolfram Language code: g = [image];
Wolfram Language code: components = ConnectedComponents[g]

Construct the condensation of g by finding the edges between the components:

Wolfram Language code: edges = Union[Cases[EdgeList[g] /. Table[v -> First[Select[components, MemberQ[#, v]&]], {v, VertexList[g]}], u_v_ /; u =!= v]];
Wolfram Language code: cond = Graph[components, edges]

Use the topological ordering of the strongly connected components to order the vertices of g:

Wolfram Language code: TopologicalSort[cond]
Wolfram Language code: h = Graph[Join@@%, EdgeList[g], Options[g]]

The new adjacency matrix is block upper triangular:

Wolfram Language code: ArrayPlot /@ {AdjacencyMatrix[g], AdjacencyMatrix[h]}
Wolfram Research (2010), TopologicalSort, Wolfram Language function, https://reference.wolfram.com/language/ref/TopologicalSort.html (updated 2015).

Text

Wolfram Research (2010), TopologicalSort, Wolfram Language function, https://reference.wolfram.com/language/ref/TopologicalSort.html (updated 2015).

CMS

Wolfram Language. 2010. "TopologicalSort." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/TopologicalSort.html.

APA

Wolfram Language. (2010). TopologicalSort. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/TopologicalSort.html

BibTeX

@misc{reference.wolfram_2026_topologicalsort, author="Wolfram Research", title="{TopologicalSort}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/TopologicalSort.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_topologicalsort, organization={Wolfram Research}, title={TopologicalSort}, year={2015}, url={https://reference.wolfram.com/language/ref/TopologicalSort.html}, note=[Accessed: 13-June-2026]}