# On Visibility Representation of Plane Graphs

Ontology type: schema:Chapter      Open Access: True

### Chapter Info

DATE

2004

AUTHORS ABSTRACT

In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan [11], Tamassia and Tollis [14] independently gave linear time VR algorithms for 2-connected plane graph. Recently, Lin et. al. reduced the width bound to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lfloor \frac{22n - 42}{15} \rfloor$\end{document} [10]. In this paper, we prove that any plane graph G has a VR with width at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lfloor \frac{13n - 24}{9} \rfloor$\end{document}.For a 4-connected plane triangulation G, we give a visibility representation of G with height at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lceil \frac{3n}{4} \rceil$\end{document}. In order to show that, we first show that every such graph has a canonical ordering tree with at most \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lceil \frac{n+1}{2} \rceil$\end{document} leaves instead of the previously known bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lfloor \frac{2n + 1}{3} \rfloor$\end{document}, which is of independent interest. All of them can be obtained in linear time. More... »

PAGES

477-488

### Book

TITLE

STACS 2004

ISBN

978-3-540-21236-2
978-3-540-24749-4

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-540-24749-4_42

DOI

http://dx.doi.org/10.1007/978-3-540-24749-4_42

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1036759274

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