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Track Layouts, Layered Path Decompositions, and Leveled Planarity

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Abstract

We investigate two types of graph layouts, track layouts and layered path decompositions, and the relations between their associated parameters track-number and layered pathwidth. We use these two types of layouts to characterize leveled planar graphs, which are the graphs with planar leveled drawings with no dummy vertices. It follows from the known NP-completeness of leveled planarity that track-number and layered pathwidth are also NP-complete, even for the smallest constant parameter values that make these parameters nontrivial. We prove that the graphs with bounded layered pathwidth include outerplanar graphs, Halin graphs, and squaregraphs, but that (despite having bounded track-number) series–parallel graphs do not have bounded layered pathwidth. Finally, we investigate the parameterized complexity of these layouts, showing that past methods used for book layouts do not work to parameterize the problem by treewidth or almost-tree number but that the problem is (non-uniformly) fixed-parameter tractable for tree-depth.

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Authors and Affiliations

  1. Pinterest, 808 Brannan Street, San Francisco, CA, 94103, USA

    Michael J. Bannister

  2. Department of Computer Science, University of California, Irvine, CA, USA

    William E. Devanny & David Eppstein

  3. School of Computer Science and Electrical Engineering, University of Ottawa, Ottawa, Canada

    Vida Dujmović

  4. School of Mathematical Sciences, Monash University, Melbourne, Australia

    David R. Wood

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  1. Michael J. Bannister
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  4. David Eppstein
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Correspondence to David R. Wood.

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A preliminary version of this paper entitled “Track Layout is Hard” was published in Proc. of 24th International Symp. on Graph Drawing and Network Visualization (GD ’16), Lecture Notes in Computer Science 9801:499–510, Springer, 2016.

Michael J. Bannister and David Eppstein were supported in part by NSF Grant CCF-1228639. William E. Devanny was supported by an NSF Graduate Research Fellowship under Grant DGE-1321846. Vida Dujmović was supported by NSERC and the Ministry of Research and Innovation, Government of Ontario, Canada. David R. Wood was supported by the Australian Research Council.

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Bannister, M.J., Devanny, W.E., Dujmović, V. et al. Track Layouts, Layered Path Decompositions, and Leveled Planarity. Algorithmica 81, 1561–1583 (2019). https://doi.org/10.1007/s00453-018-0487-5

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