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In [[topological graph theory]], an '''embedding''' (also spelled '''imbedding''') of a [[Graph (discrete mathematics)|graph]] <math>G</math> on a [[surface (mathematics)|surface]] <math>\Sigma</math> is a representation of <math>G</math> on <math>\Sigma</math> in which points of <math>\Sigma</math> are associated with [[graph theory|vertices]] and simple arcs ([[Homeomorphism|homeomorphic]] images of <math>[0,1]</math>) are associated with [[graph theory|edges]] in such a way that:
* the endpoints of the arc associated with an edge <math>e</math> are the points associated with the end vertices of <math>e</math>‍,
* no arcs include points associated with other vertices,
* two arcs never intersect at a point which is interior to either of the arcs.
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