Re: "Embedded" in the plane.

From: W. Dale Hall (mailtowd_hall_at_pacbell.net)
Date: 08/01/04 

Date: Sun, 01 Aug 2004 09:06:07 GMT

Bill Jones wrote:

> David Eppstein <eppstein@ics.uci.edu> wrote in message news:<eppstein-AC8FF4.18014530072004@news.service.uci.edu>...
>
>
>>Bill Jones ... naive-appearing question.
>
>
> I have another "naive" qestion.
>
> Does "embedded in the plane" refer to all graphs or just to planar graphs?

It refers to all topological spaces (at least). However, the question

        Can X be embedded in the plane?

usually gives the answer "No".

When restricted to graphs (in some ways of understanding, a graph can be
considered a topological space), then the answer is "Yes" for planar
graphs.

That's why they're called "planar". Because they *can* be embedded in
the plane. Note, however, that some depictions of a planar graph
as an arrangement of points and arcs in the plane are not embeddings:

            NOT AN EMBEDDING AN EMBEDDING:

             o-----------o o-----------o
              \ / | |
               \ / | |
                \ / | |
                 \ / | |
                  \ / | |
                   X <---distinct o-----------o
                  / \ edges meet
                 / \
                / \
               / \
              / \
             o-----------o

Dale.

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