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International Conference on Foundations of Software Technology and Theoretical Computer Science

FSTTCS 2002: FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science pp 348–359Cite as

Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2556))

Abstract

A three-dimensional (straight-line grid) drawing of a graph represents the vertices by points in Z3 and the edges by non-crossing line segments. This research is motivated by the following open problem due to Felsner, Liotta, and Wismath [Graph Drawing ’01, Lecture Notes in Comput. Sci., 2002]: does every n-vertex planar graph have a threedimensional drawing with O(n) volume? We prove that this question is almost equivalent to an existing one-dimensional graph layout problem. A queue layout consists of a linear order σ of the vertices of a graph, and a partition of the edges into queues, such that no two edges in the same queue are nested with respect to σ. The minimum number of queues in a queue layout of a graph is its queue-number. Let G be an n-vertex member of a proper minor-closed family of graphs (such as a planar graph). We prove that G has a O(1) × O(1) × O(n) drawing if and only if G has O(1) queue-number. Thus the above question is almost equivalent to an open problem of Heath, Leighton, and Rosenberg [SIAM J. Discrete Math., 1992], who ask whether every planar graph has O(1) queue-number? We also present partial solutions to an open problem of Ganley and Heath [Discrete Appl. Math., 2001], who ask whether graphs of bounded tree-width have bounded queue-number? We prove that graphs with bounded path-width, or both bounded tree-width and bounded maximum degree, have bounded queue-number. As a corollary we obtain three-dimensional drawings with optimal O(n) volume, for series-parallel graphs, and graphs with both bounded tree-width and bounded maximum degree.

Research supported by NSERC.

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Authors and Affiliations

  1. School of Computer Science, Carleton University, Ottawa, Canada

    David R. Wood

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  1. David R. Wood
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Editors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology, 208016, Kanpur, India

    Manindra Agrawal  & Anil Seth  & 

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Wood, D.R. (2002). Queue Layouts, Tree-Width, and Three-Dimensional Graph Drawing. In: Agrawal, M., Seth, A. (eds) FST TCS 2002: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2002. Lecture Notes in Computer Science, vol 2556. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36206-1_31

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  • DOI: https://doi.org/10.1007/3-540-36206-1_31

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  • Print ISBN: 978-3-540-00225-3

  • Online ISBN: 978-3-540-36206-7

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