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PlanarGraphQ[g]

yields True if g is a planar graph and False otherwise.

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Scope  
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PlanarGraphQ

PlanarGraphQ[g]

yields True if g is a planar graph and False otherwise.

Details

  • A graph is planar if it can be drawn in the plane without edge crossings.

Examples

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Basic Examples  (2)

Test whether a graph is planar:

Wolfram Language code: GraphData[{"Antiprism", 4}]
Wolfram Language code: PlanarGraphQ[%]

Not all graphs are planar:

Wolfram Language code: PetersenGraph[5, 2]
Wolfram Language code: PlanarGraphQ[%]

Scope  (6)

PlanarGraphQ works with undirected graphs:

Wolfram Language code: PlanarGraphQ[[image]]

Directed graphs:

Wolfram Language code: PlanarGraphQ[[image]]

Multigraphs:

Wolfram Language code: PlanarGraphQ[[image]]

Mixed graphs:

Wolfram Language code: PlanarGraphQ[[image]]

PlanarGraphQ gives False for expressions that are not graphs:

Wolfram Language code: PlanarGraphQ[x]

PlanarGraphQ works with large graphs:

Wolfram Language code: GridGraph[{10, 10, 10, 10}];
Wolfram Language code: PlanarGraphQ[%]//Timing

Applications  (2)

Four colors can be used to color a map with no two adjacent countries having the same color:

Wolfram Language code: PlanarGraphQ[\!\(\*GraphicsBox[«4»]\)]

A color scheme:

Wolfram Language code: colors = {Red, Yellow, Blue, Green}[[{4, 2, 1, 3, 3, 1, 2, 2, 4, 3, 4, 3, 2, 2}]];
Wolfram Language code: Graphics[{EdgeForm[Black], MapIndexed[{colors[[First[#2]]], CountryData[#, "Polygon"]}&, CountryData["SouthAmerica"]]}]

Check whether non-overlapping utility lines can be used to distribute gas, water, and electricity:

Wolfram Language code: PlanarGraphQ[[image]]

Properties & Relations  (3)

Use GraphData for an extensive collection of planar graphs:

Wolfram Language code: Short[GraphData["Planar"]]

Non-planar graphs:

Wolfram Language code: Short[GraphData["Nonplanar"]]

Use GraphLayout->"PlanarLayout" to lay out a planar graph:

Wolfram Language code: HararyGraph[4, 8, GraphLayout -> "PlanarLayout"]

For connected planar graphs with vertices and edges, with the number of faces:

Wolfram Language code: g = HypercubeGraph[3]
Wolfram Language code: {ConnectedGraphQ[g], PlanarGraphQ[g]}
Wolfram Language code: VertexCount[g] - EdgeCount[g] + 6
Wolfram Research (2012), PlanarGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanarGraphQ.html.

Text

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

CMS

Wolfram Language. 2012. "PlanarGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PlanarGraphQ.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_planargraphq, author="Wolfram Research", title="{PlanarGraphQ}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/PlanarGraphQ.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_planargraphq, organization={Wolfram Research}, title={PlanarGraphQ}, year={2012}, url={https://reference.wolfram.com/language/ref/PlanarGraphQ.html}, note=[Accessed: 13-June-2026]}