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.
