A planar graph is an undirected graph that can be drawn in the plane in such a way that no two edges intersect except at a common vertex. The drawing is called a Plane Graph
- A map is a -connected plane graph.
The plane graph will divide the plane into regions called faces. The unbounded region on the exterior of the plane graph is called the infinite face.
The crossing number of , denoted is the minimum number of crossings that can occur when is drawn in the plane
The thickness of a graph is the number of planar graphs such that the union of these planar graphs is . This is denoted
A graph is outerplanar if it can be drawn in the plane so that all vertices lie on the exterior boundary.

Topological Extensions

A graph is of genus if it can be drawn on a surface with genus but not on one of genus
Let be graphs. They are homeomorphic if they are isomorphic to each other after performing either edge subdivision or edge smoothing. That is, they look the same when their plane graphs are drawn.

Ringel-Youngs Theorem
(Wilson 14.1) The genus of a graph does not exceed its crossing number.
(Wilson 14.2) Let be a connected graph of genus with vertices, edges and faces. Then

This is a variation of Euler’s Formula
(Wilson 14.3) The genus of a simple graph of vertices and edges satisfies
(Wilson 14.3.z ) If the graph has girth , then the genus satisfies

Theorems

Let be a plane graph of a connected planar graph with vertices edges and faces.

We have
An informal way to see this: every edge either introduces a new face or joins with a lone vertex.
(Wilson 13.3) Let be the number of components. Then:

Every Planar Graph can be drawn such that all the edges in the Plane graph are line segments

A graph is planar if and only if it contains no subgraph that is homeomorphic to either or
Outerplanar Extension a graph is outerplanar if and only if contains no subgraph homeomorphic or edge contractible to or .
(Wilson 12.3) A graph is planar if and only if it contains no subgraph contractible to either or

(Wilson 13.4a) If is a connected simple planar graph with vertices and edges, then
(Wilson 13.6) Every simple planar graph has a vertex with minimum degree .
(Wilson 13.7) For vertices, the thickness satisfies the following:
(Wilson e13.3) ¹ If is a connected, simple, planar graph with girth , vertices and edges, then
(Wilson 15.6) A graph is planar if and only if it has an algebraic dual.
(Wilson 15.9) Let be a connected, plane graph. is bipartite if and only if is Eulerian.