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Algorithmica

pp 1–23 | Cite as

Track Layouts, Layered Path Decompositions, and Leveled Planarity

  • Michael J. Bannister
  • William E. Devanny
  • Vida Dujmović
  • David Eppstein
  • David R. Wood
Article

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.

Keywords

Track layouts Layered path decompositions Track-number Layered pathwidth Leveled planar graphs Outerplanar graphs Halin graphs Squaregraphs Unit disc graphs Parameterized complexity Treewidth Almost-tree number Tree-depth 

Notes

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.PinterestSan FranciscoUSA
  2. 2.Department of Computer ScienceUniversity of CaliforniaIrvineUSA
  3. 3.School of Computer Science and Electrical EngineeringUniversity of OttawaOttawaCanada
  4. 4.School of Mathematical SciencesMonash UniversityMelbourneAustralia

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