Graph embedding

In topological graph theory, an embedding (also spelled imbedding) of a graph ${\displaystyle G}$ on a surface Σ is a representation of ${\displaystyle G}$ on Σ in which points of Σ are associated with vertices and simple arcs (homeomorphic images of [0,1]) are associated with edges in such a way that:

Here a surface is a compact, connected 2-manifold.

Informally, an embedding of a graph into a surface is a drawing of the graph on the surface in such a way that its edges may intersect only at their endpoints. It is well known that any finite graph can be embedded in 3-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{3}}$ and planar graphs can be embedded in 2-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{2}}$.

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