Graph embedding
Embedding a graph in a topological space, often Euclidean / From Wikipedia, the free encyclopedia
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In topological graph theory, an embedding (also spelled imbedding) of a graph on a surface is a representation of on in which points of are associated with vertices and simple arcs (homeomorphic images of ) are associated with edges in such a way that:
- the endpoints of the arc associated with an edge are the points associated with the end vertices of
- 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 -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 .[1] A planar graph is one that can be embedded in 2-dimensional Euclidean space
Often, an embedding is regarded as an equivalence class (under homeomorphisms of ) 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".