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# Planar Embedding

A planar embedding, also called a "plane graph" (Harary 1994, p. 103; Harborth and Möller 1994), "planar drawing," or "plane drawing," of a planar graph is an embedding in which no two edges intersect (or overlap) and no two vertices coincide. Equivalently, a planar embedding is an embedding of a graph drawn in the plane such that edges intersect only at their endpoints.

A planar straight line embedding of a planar graph can be constructed in the Wolfram Language using the "PlanarEmbedding" option to GraphLayout or using PlanarGraph[g].

Precomputed planar embeddings of some graphs are available in the Wolfram Language as GraphData[g, "Graph", "Planar"].

In general, planar graphs may have multiple homeomorphically distinct planar embeddings on the sphere. Graphs with a single homeomorphically distinct planar embedding are called uniquely embeddable, among which are all polyhedral graphs. Uniquely embeddable graphs have a unique dual graph.

The numbers of embeddings on the sphere of 2-connected planar graphs with , 2, ... nodes are given by 0, 0, 1, 3, 10, 61, 564, 7593, 123874, ... (OEIS A034889). The first case where this exceeds the number of nonisomorphic 2-connected planar graphs occurs for , where a single 5-vertex planat graph has two distinct planar embeddings on the sphere.

Facially Complete Planar Embedding, Graph Embedding, Planar Graph, Planar Straight Line Embedding, Uniquely Embeddable

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## References

Harary, F. Graph Theory. Reading, MA: Addison-Wesley, 1994.Harborth, H. and Möller, M. "Minimum Integral Drawings of the Platonic Graphs." Math. Mag. 67, 355-358, 1994.Sloane, N. J. A. Sequence A034889 in "The On-Line Encyclopedia of Integer Sequences."

## Cite this as:

Weisstein, Eric W. "Planar Embedding." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PlanarEmbedding.html