Graph embedding

Idea
Informally an embedding is a way to draw a graph on a surface without edges intersecting.

Definition
The embedding of a graph $G$ on a connected, manifold $$\Sigma$$ consists of two mappings. The first mapping $$\psi$$ the vertices of $G$ to points on $$\Sigma$$ injectively, that is no two vertices map to the same point. The second mapping, $$\rho$$, maps the edges of $G$ to arcs on the surface of $$\Sigma$$. An arc is a subset of $$\Sigma$$ which is homeomorphic to $$[0,1]\subseteq\mathbb{R}$$.

The two mappings are required


 * If vertex $v$ is incident on edge $e$, then $$\psi(v)$$ is an endpoint of $$\rho(e)$$, that is there is a homeomorphism $$\phi : [0,1]\rightarrow \rho(e)$$ such that $$\phi(0) = \psi(v)$$.
 * If vertex $v$ is not incident on edge $e$, then $$\psi(v)\notin\rho(v)$$.
 * If $$e_0 \neq e_1$$ then $$\rho(e_0)$$ and $$\rho(e_1)$$ do not intersect anywhere other than their endpoints.

Planar graphs
A graph is planar iff there is an embedding of it onto the plane.