Optimal st-Orientations for Plane Triangulations View Full Text


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Chapter Info

DATE

2007-01-01

AUTHORS

Huaming Zhang , Xin He

ABSTRACT

For a plane triangulation G with n vertices, it has been proved that there exists a plane triangulation G with n vertices such that for any st-orientation of G, the length of the longest directed paths of G from s to t is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\geq \lfloor \frac{2n}{3}\rfloor$\end{document} [18] . In this paper, we prove the bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac{2n}{3}$\end{document} is optimal by showing that every plane triangulation G with n-vertices admits an st-orientation with the length of its longest directed paths bounded by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\frac {2n}{3}+O(1)$\end{document}. In addition, this st-orientation is constructible in linear time. A by-product of this result is that every plane graph G with n vertices admits a visibility representation with height \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\le \frac{2n}{3}+O(1)$\end{document}, constructible in linear time, which is also optimal. More... »

PAGES

296-305

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-540-72870-2_28

DOI

http://dx.doi.org/10.1007/978-3-540-72870-2_28

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1022421972


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