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PlanarFaceList[g]

gives the list of faces of the planar graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
IncludeOuterFace  
Applications  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
Cite this Page

PlanarFaceList

PlanarFaceList[g]

gives the list of faces of the planar graph g.

Details and Options

  • PlanarFaceList is also known as faces or regions.
  • PlanarFaceList is typically used to give vertices of the dual of a planar graph.
  • PlanarFaceList[g] gives a list of faces {f1,f2,…} of the graph g, where the face fi={vi1,vi2,…} is bounded by vertices vij and edges between them.
  • With the setting IncludeOuterFace False, the outer face is excluded from the result.

Examples

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Basic Examples  (2)

The list of faces of a planar graph:

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

The three-dimensional HypercubeGraph is planar:

Wolfram Language code: PlanarGraph[HypercubeGraph[3]]
Wolfram Language code: PlanarFaceList[%]

Scope  (4)

PlanarFaceList works with undirected graphs:

Wolfram Language code: PlanarFaceList[[image]]

Directed graphs:

Wolfram Language code: PlanarFaceList[[image]]

Multigraphs:

Wolfram Language code: PlanarFaceList[[image]]

Mixed graphs:

Wolfram Language code: PlanarFaceList[[image]]

Options  (1)

IncludeOuterFace  (1)

By default, the outer face is included:

Wolfram Language code: PlanarFaceList[HypercubeGraph[3]]

Exclude the outer face:

Wolfram Language code: PlanarFaceList[HypercubeGraph[3], IncludeOuterFace -> False]

Applications  (1)

Color the graph of a trabecular bone structure:

Wolfram Language code: i = [image];
Wolfram Language code: b = ColorNegate[Image[WatershedComponents[i]]];
Wolfram Language code: g = MorphologicalGraph[b]

Find faces and corresponding colors:

Wolfram Language code: faces = PlanarFaceList[g];
Wolfram Language code: colors = FindPlanarColoring[g, ColorData[106, "ColorList"]];

Color faces:

Wolfram Language code: coords = AssociationThread[VertexList[g] -> GraphEmbedding[g]];
Wolfram Language code: Graphics[Table[Style[Polygon[faces[[i]] /. coords], colors[[i]]], {i, Length[colors]}]]

Properties & Relations  (2)

Use DualPlanarGraph to get the dual of planar graphs:

Wolfram Language code: DualPlanarGraph[[image]]

Use FindPlanarColoring to find a coloring for the faces of planar graphs:

Wolfram Language code: FindPlanarColoring[[image]]

Possible Issues  (3)

PlanarFaceList returns faces based on the given layout:

Wolfram Language code: Graph[[image], GraphLayout -> "PlanarEmbedding"]
Wolfram Language code: PlanarFaceList[%]

PlanarFaceList returns faces based on the given coordinates:

Wolfram Language code: Graph[[image], VertexCoordinates -> {{0, 0}, {3, 0}, {1, 1}, {0, 3}}]
Wolfram Language code: PlanarFaceList[%]

PlanarFaceList can return non-simple faces:

Wolfram Language code: PlanarFaceList[[image]]
Wolfram Research (2021), PlanarFaceList, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanarFaceList.html.

Text

Wolfram Research (2021), PlanarFaceList, Wolfram Language function, https://reference.wolfram.com/language/ref/PlanarFaceList.html.

CMS

Wolfram Language. 2021. "PlanarFaceList." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/PlanarFaceList.html.

APA

Wolfram Language. (2021). PlanarFaceList. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/PlanarFaceList.html

BibTeX

@misc{reference.wolfram_2026_planarfacelist, author="Wolfram Research", title="{PlanarFaceList}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/PlanarFaceList.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_planarfacelist, organization={Wolfram Research}, title={PlanarFaceList}, year={2021}, url={https://reference.wolfram.com/language/ref/PlanarFaceList.html}, note=[Accessed: 12-June-2026]}