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clarify — intent was finite graphs only (although it also works for infinite simple graphs with at most a continuum number of vertices) |
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Here a surface is a [[compact space|compact]], [[connected space|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 <math>\mathbb{R}^3</math><ref name="3d-gd">{{citation
| last1 = Cohen | first1 = Robert F.
| last2 = Eades | first2 = Peter | author2-link = Peter Eades
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==Embeddings of graphs into higher-dimensional spaces==
It is known that any finite graph can be embedded into a three-dimensional space.<ref name="3d-gd"/>
One method for doing this is to place the points on any line in space and to draw the
Alternatively, any finite graph can be drawn with straight-line edges in three dimensions without crossings by placing its vertices in [[general position]] so that no four are coplanar. For instance, this may be achieved by placing the ''i''th vertex at the point (''i'',''i''<sup>2</sup>,''i''<sup>3</sup>) of the [[moment curve]].
An embedding of a graph into three-dimensional space in which no two of the cycles are topologically linked is called a [[linkless embedding]]. A graph has a linkless embedding if and only if it does not have one of the seven graphs of the [[Petersen family]] as a [[minor (graph theory)|minor]].
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