# Visibility Representations of Four-Connected Plane Graphs with Near Optimal Heights

Ontology type: schema:Chapter      Open Access: True

### Chapter Info

DATE

2009

AUTHORS ABSTRACT

A visibility representation of a graph G is to represent the nodes of G with non-overlapping horizontal line segments such that the line segments representing any two distinct adjacent nodes are vertically visible to each other. If G is a plane graph, i.e., a planar graph equipped with a planar embedding, a visibility representation of G has the additional requirement of reflecting the given planar embedding of G. For the case that G is an n-node four-connected plane graph, we give an O(n)-time algorithm to produce 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} $\left\lceil\frac{n}{2}\right\rceil+2\left\lceil\sqrt{\frac{n-2}{2}}\right\rceil$ \end{document}. To ensure that the first-order term of the upper bound is optimal, we also show an n-node four-connected plane graph G, for infinite number of n, whose visibility representations require heights at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac{n}{2}$ \end{document}. More... »

PAGES

67-77

### Book

TITLE

Graph Drawing

ISBN

978-3-642-00218-2
978-3-642-00219-9

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-00219-9_8

DOI

http://dx.doi.org/10.1007/978-3-642-00219-9_8

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1047130880

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