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On the Parameterized Complexity of Layered Graph Drawing

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Abstract

We consider graph drawings in which vertices are assigned to layers and edges are drawn as straight line-segments between vertices on adjacent layers. We prove that graphs admitting crossing-free h-layer drawings (for fixed h) have bounded pathwidth. We then use a path decomposition as the basis for a linear-time algorithm to decide if a graph has a crossing-free h-layer drawing (for fixed h). This algorithm is extended to solve related problems, including allowing at most k crossings, or removing at most r edges to leave a crossing-free drawing (for fixed k or r). If the number of crossings or deleted edges is a non-fixed parameter then these problems are NP-complete. For each setting, we can also permit downward drawings of directed graphs and drawings in which edges may span multiple layers, in which case either the total span or the maximum span of edges can be minimized. In contrast to the Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, Montreal, Quebec, H3A 2T5, Canada

    Vida Dujmović, Matthew Kitching & Sue Whitesides

  2. School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, Australia

    Michael R. Fellows & Frances Rosamond

  3. Dipartimento di Ingegneria Elettronica e dell’Informazione, Università degli Studi di Perugia, Perugia, Italy

    Giuseppe Liotta

  4. Institute of Information Science and Technology, Massey University, Palmerston North, New Zealand

    Catherine McCartin

  5. School of Computer Science, University of Waterloo, Waterloo, Ontario, N2L 2P9, Canada

    Naomi Nishimura & Prabhakar Ragde

  6. Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

    David R. Wood

Authors
  1. Vida Dujmović
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  2. Michael R. Fellows
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  3. Matthew Kitching
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  4. Giuseppe Liotta
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  5. Catherine McCartin
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  6. Naomi Nishimura
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  7. Prabhakar Ragde
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  8. Frances Rosamond
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  9. Sue Whitesides
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  10. David R. Wood
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Corresponding author

Correspondence to Prabhakar Ragde.

Additional information

Research initiated at the International Workshop on Fixed Parameter Tractability in Graph Drawing, Bellairs Research Institute of McGill University, Holetown, Barbados, Feb. 9–16, 2001, organized by S. Whitesides. Research of Canada-based authors is supported by NSERC; research of Quebec-based authors is also supported by a grant from FCAR. Research of D.R. Wood completed while visiting McGill University. Research of G. Liotta supported by CNR and MURST.

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Dujmović, V., Fellows, M.R., Kitching, M. et al. On the Parameterized Complexity of Layered Graph Drawing. Algorithmica 52, 267–292 (2008). https://doi.org/10.1007/s00453-007-9151-1

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  • DOI: https://doi.org/10.1007/s00453-007-9151-1

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