# On Open Rectangle-of-Influence Drawings of Planar Graphs

Ontology type: schema:Chapter

### Chapter Info

DATE

2009

AUTHORS ABSTRACT

We investigate open rectangle-of-influence drawings for irreducible triangulations, which are plane graphs with a quadrangular exterior face, triangular interior faces and no separating triangles. An open rectangle-of-influence drawing of a plane graph G is a type of straight-line grid drawing where there is no vertices drawn in the proper inside of the axis-parallel rectangle defined by the two end vertices of any edge. The algorithm presented by Miura and Nishizeki [8] uses a grid of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal W} + {\cal H} \leq$\end{document}(n-1), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal W}$\end{document} is the width of the grid, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\cal H}$\end{document} is the height of the grid and n is the number of vertices in G. Thus the area of the grid is at most ⌈(n-1)/2⌉ × \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lfloor$\end{document}(n-1)/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\rfloor$\end{document} [8].In this paper, we prove that the two straight-line grid drawing algorithms for irreducible triangulations from [4] and quadrangulations from [3,5] actually produce open rectangle-of-influence drawings for them respectively. Therefore, the straight-line grid drawing size bounds from [3,4,5] also hold for the open rectangle-of-influence drawings. For irreducible triangulations, the new asymptotical grid size bound is \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\emph{11n/27}$\end{document} × \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\emph{11n/27}$\end{document}. For quadrangulations, our asymptotical grid size bound \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\emph{13n/27}$\end{document} × \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\emph{13n/27}$\end{document} is the first known such bound. More... »

PAGES

123-134

### Book

TITLE

Combinatorial Optimization and Applications

ISBN

978-3-642-02025-4
978-3-642-02026-1

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/978-3-642-02026-1_11

DOI

http://dx.doi.org/10.1007/978-3-642-02026-1_11

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1052006689

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