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Graph genus

In topological graph theory, an embedding (also spelled imbedding) of a graphG{\displaystyle G} on a surfaceΣ{\displaystyle \Sigma } is a representation of G{\displaystyle G} on Σ{\displaystyle \Sigma } in which points of Σ{\displaystyle \Sigma } are associated with vertices and simple arcs (homeomorphic images of [0,1]{\displaystyle [0,1]}) are associated with edges in such a way that: the endpoints of the arc associated with an edge e{\displaystyle e} are the points associated with the end vertices of e,{\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 connected2{\displaystyle 2}-manifold.

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