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PathGraphQ[g]

yields True if the graph g is a path and False otherwise.

Details
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Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
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PathGraphQ

PathGraphQ[g]

yields True if the graph g is a path and False otherwise.

Details

  • An undirected path graph is a connected graph where each vertex has at most degree two.
  • A directed path graph is a connected graph where each vertex has at most in-degree one and at most out-degree one.

Examples

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Basic Examples  (2)

Test whether a graph is a path:

Wolfram Language code: Graph[{12, 23, 34}]
Wolfram Language code: PathGraphQ[%]

The vertex degree is at most 2:

Wolfram Language code: VertexDegree[%%]

A complete graph is not a path:

Wolfram Language code: CompleteGraph[5]
Wolfram Language code: PathGraphQ[%]

The vertex degree is greater than 2:

Wolfram Language code: VertexDegree[%%]

Scope  (6)

PathGraphQ works with undirected graphs:

Wolfram Language code: PathGraphQ[[image]]

Directed graphs:

Wolfram Language code: PathGraphQ[[image]]

Multigraphs:

Wolfram Language code: PathGraphQ[[image]]

Mixed graphs:

Wolfram Language code: PathGraphQ[[image]]

PathGraphQ gives False for anything that is not a path graph:

Wolfram Language code: PathGraphQ[x]
Wolfram Language code: PathGraphQ[Graph[garbage]]

Test large graphs:

Wolfram Language code: g = GridGraph[{1, 10000}];
Wolfram Language code: PathGraphQ[g]//Timing
Wolfram Language code: {VertexCount[g], EdgeCount[g]}

Properties & Relations  (8)

A path graph is loop-free if it has more than one vertex:

Wolfram Language code: Graph[{12, 23, 22}]
Wolfram Language code: {PathGraphQ[%], LoopFreeGraphQ[%]}

A path graph does not necessarily have edges:

Wolfram Language code: PathGraphQ[Graph[{1}, {}]]

A path graph that starts and ends in the same vertex is a cycle graph:

Wolfram Language code: CycleGraph[5]
Wolfram Language code: PathGraphQ[%]

A path graph with no repeated vertices is a tree:

Wolfram Language code: g = Graph[{12, 23, 34}];
Wolfram Language code: {TreeGraphQ[g], PathGraphQ[g]}

An acyclic path graph is simple:

Wolfram Language code: g = GridGraph[{1, 4}]
Wolfram Language code: {PathGraphQ[g] && AcyclicGraphQ[g], SimpleGraphQ[g]}

And also bipartite:

Wolfram Language code: BipartiteGraphQ[g]

GridGraph[{1,…,1,k,1,…,1}] are all path graphs:

Wolfram Language code: PathGraphQ[GridGraph[{1, 4}]]
Wolfram Language code: PathGraphQ[GridGraph[{4, 1}]]
Wolfram Language code: PathGraphQ[GridGraph[{1, 4, 1}]]

A path graph is connected and each vertex has at most degree 2:

Wolfram Language code: g = CycleGraph[5]
Wolfram Language code: {ConnectedGraphQ[g], PathGraphQ[g]}
Wolfram Language code: Max[VertexDegree[g]]

The line graph of a path is isomorphic to :

Wolfram Language code: PathGraph[Range[8]]
Wolfram Language code: IsomorphicGraphQ[LineGraph[%], PathGraph[Range[7]]]

Possible Issues  (1)

PathGraphQ gives False for non-explicit graphs:

Wolfram Language code: PathGraphQ[CycleGraph[n]]
Wolfram Language code: PathGraphQ[CompleteGraph[n]]
Wolfram Research (2010), PathGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PathGraphQ.html.

Text

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

CMS

Wolfram Language. 2010. "PathGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PathGraphQ.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_pathgraphq, author="Wolfram Research", title="{PathGraphQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/PathGraphQ.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_pathgraphq, organization={Wolfram Research}, title={PathGraphQ}, year={2010}, url={https://reference.wolfram.com/language/ref/PathGraphQ.html}, note=[Accessed: 12-June-2026]}