Graph embedding

Article summary

and associated map embedded in the torus. In topological graph theory, an embedding (also spelled imbedding) of a graph G on a surface \Sigma is a representation of G on \Sigma in which points of \Sigma are associated with vertices and simple arcs (homeomorphic images of [0,1] ) are associated with edges in such a way that: * the endpoints of the arc associated with an edge e are the points associated with the end vertices of 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 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 \mathbb{R}^3 . A planar graph is one that can be embedded in 2-dimensional Euclidean space \mathbb{R}^2. Often, an embedding is regarded as an equivalence class (under homeomorphisms of \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". This article deals only with the strict definition of graph embedding.…

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