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 a large number of 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 the total span or the maximum span of edges can be minimized. In contrast to the so-called Sugiyama method for layered graph drawing, our algorithms do not assume a preassignment of the vertices to layers.
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. Contact author: P. Ragde, Dept. of Computer Science, University of Waterloo, Waterloo, Ontario, Canada N2L 2P9,e-mail plragde@uwaterloo.ca. Research of Canada-based authors is supported byNSERC. Research of D. R. Wood supported by ARC and completed while visiting McGill University. Research of G. Liotta supported by CNR and MURST.
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Dujmović, V. et al. (2001). On the Parameterized Complexity of Layered Graph Drawing. In: auf der Heide, F.M. (eds) Algorithms — ESA 2001. ESA 2001. Lecture Notes in Computer Science, vol 2161. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44676-1_41
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DOI: https://doi.org/10.1007/3-540-44676-1_41
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