A polynomial bound for untangling geometric planar graphs

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Document typeArticle
Defense date2009-12
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Abstract
To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos (Discrete Comput. Geom. 28(4): 585–592, 2002) asked if every n-vertex geometric planar graph can be untangled while keeping at least $n^\in{}$ vertices fixed. We answer this question in the affirmative with ∊ = 1/4. The previous best known bound was Ω$(\sqrt{log\,n/log\,log\,n})$. We also consider untangling geometric trees. It is known that every n-vertex geometric tree can be untangled while keeping at least $(\sqrt{n/3})$ vertices fixed, while the best upper bound was O$((n\,log\,n)^{2/3})$. We answer a question of Spillner and Wolff
(http://arxiv.org/abs/0709.0170) by closing this gap for untangling trees. In particular,
we show that for infinitely many values of n, there is an n-vertex geometric tree that
cannot be untangled while keeping more than $3(\sqrt{n}-1)$ vertices fixed.
CitationBose, P. [et al.]. A polynomial bound for untangling geometric planar graphs. "Discrete and computational geometry", Desembre 2009, vol. 42, núm. 4, p. 570-585.
ISSN0179-5376
Publisher versionhttp://arxiv.org/PS_cache/arxiv/pdf/0710/0710.1641v2.pdf
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