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Handbook of Graph Drawing and Visualization
book

Handbook of Graph Drawing and Visualization

by Roberto Tamassia
August 2013
Intermediate to advanced content levelIntermediate to advanced
862 pages
32h 20m
English
Chapman and Hall/CRC
Content preview from Handbook of Graph Drawing and Visualization
48 CHAPTER 2. CROSSINGS AND PLANARIZATION
In 1983, Leighton used induction on the number of vertices to show the following theorem;
see [Lei83].
Theorem 2.1 Let G = (V, E) be a simple graph. If |E| 4|V |, we have
cr(G)
1
100
|E|
3
|V |
2
. (2.3)
Ajtai et al. obtained the same result independently with a smaller constant of
1
375
in
[ACNS82]. One of the best-known results has been derived by Pach and T´oth [PT97]. For
any simple graph G = (V, E), cr(G) satisfies
cr(G)
1
33.75
|E|
3
|V |
2
0.9|V | . (2.4)
Apart from bounds with respect to the number of vertices and edges, several approaches to
obtain tight lower bounds base d on different graph properties can be
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Publisher Resources

ISBN: 9781420010268