Graph embedding

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

• the endpoints of the arc associated with an edge ${\displaystyle e}$ are the points associated with the end vertices of ${\displaystyle e,}$
• no arcs include points associated with other vertices,
• two arcs never intersect at a point which is interior to either of the arcs.

Here a surface is a connected ${\displaystyle 2}$-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 ${\displaystyle \mathbb {R} ^{3}}$.^[1] A planar graph is one that can be embedded in 2-dimensional Euclidean space ${\displaystyle \mathbb {R} ^{2}.}$

Often, an embedding is regarded as an equivalence class (under homeomorphisms of ${\displaystyle \Sigma }$) of representations of the kind just described.

Some authors define a weaker version of the definition of "graph embedding" by omitting the non-intersection condition for edges. In such contexts the stricter definition is described as "non-crossing graph embedding".^[2]

This article deals only with the strict definition of graph embedding. The weaker definition is discussed in the articles "graph drawing" and "crossing number".