On Rectilinear Duals for Vertex-Weighted Plane Graphs

• Published in: Graph Drawing Vol. 3843; pp. 61 - 72
• Main Authors: de Berg, Mark, Mumford, Elena, Speckmann, Bettina
• Format: Book Chapter Conference Proceeding
• Language: English
• Published: Berlin, Heidelberg Springer Berlin Heidelberg 2006; Springer
• Series: Lecture Notes in Computer Science
• Subjects: Applied sciences; Computer science; control theory; systems; Exact sciences and technology; Vertex(graph); Weighted graph
• ISBN: 9783540314257; 3540314253
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/11618058_6

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal G}$\end{document} = (V,E) be a plane triangulated graph where each vertex is assigned a positive weight. A rectilinear dual of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal G}$\end{document} is a partition of a rectangle into |V| simple rectilinear regions, one for each vertex, such that two regions are adjacent if and only if the corresponding vertices are connected by an edge in E. A rectilinear dual is called a cartogram if the area of each region is equal to the weight of the corresponding vertex. We show that every vertex-weighted plane triangulated graph \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathcal G}$\end{document} admits a cartogram of constant complexity, that is, a cartogram where the number of vertices of each region is constant.