Embeddings of Graphs Into Higher-dimensional Spaces
It is known that any graph can be embedded into a three-dimensional space.
One method for doing this is to place the points on any line in space and to draw the m edges as curves each of which lies in one of m distinct halfplanes having that line as their common boundary. An embedding like this in which the edges are drawn on halfplanes is called a book embedding of the graph. This metaphor comes from imagining that each of the planes where an edge is drawn is like a page of a book. It was observed that in fact several edges may be drawn in the same "page"; the book thickness of the graph is the minimum number of halfplanes needed for such a drawing.
Alternatively, any 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 ith vertex at the point (i,i2,i3) 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.
Read more about this topic: Graph Embedding
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“Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.”
—Henry David Thoreau (1817–1862)