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Layered Fan-Planar Graph Drawings

Authors Therese Biedl , Steven Chaplick , Michael Kaufmann , Fabrizio Montecchiani , Martin Nöllenburg , Chrysanthi Raftopoulou

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • Graph Drawing
  • Parameterized Complexity
  • Beyond Planar Graphs

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Abstract

In a fan-planar drawing of a graph an edge can cross only edges with a common end-vertex. In this paper, we study fan-planar drawings that use h (horizontal) layers and are proper, i.e., edges connect adjacent layers. We show that if the embedding of the graph is fixed, then testing the existence of such drawings is fixed-parameter tractable in h, via a reduction to a similar result for planar graphs by Dujmović et al. If the embedding is not fixed, then we give partial results for h = 2: It was already known how to test the existence of fan-planar proper 2-layer drawings for 2-connected graphs, and we show here how to test this for trees. Along the way, we exhibit other interesting results for graphs with a fan-planar proper h-layer drawing; in particular we bound their pathwidth and show that they have a bar-1-visibility representation.

Cite As Get BibTex

Therese Biedl, Steven Chaplick, Michael Kaufmann, Fabrizio Montecchiani, Martin Nöllenburg, and Chrysanthi Raftopoulou. Layered Fan-Planar Graph Drawings. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 14:1-14:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.14

Author Details

Therese Biedl
  • University of Waterloo, Canada
Steven Chaplick
  • Maastricht University, The Netherlands
Michael Kaufmann
  • Universität Tübingen, Germany
Fabrizio Montecchiani
  • Universitá degli Studi di Perugia, Italy
Martin Nöllenburg
  • TU Wien, Austria
Chrysanthi Raftopoulou
  • National Technical University of Athens, Greece

Funding

  • Biedl, Therese: Research supported by NSERC.
  • Chaplick, Steven: Research supported by DFG grant WO 758/11-1.
  • Montecchiani, Fabrizio: Partially supported by Dip. Ingegneria, Univ. Perugia, grant RICBA19FM.

Acknowledgements

Research initiated while Fabrizio Montecchiani was visiting the University of Waterloo and continued at the Bertinoro Workshop on Graph Drawing 2018.

References

  1. T. Biedl. Height-preserving transformations of planar graph drawings. In C. Duncan and A. Symvonis, editors, Graph Drawing (GD'14), volume 8871 of LNCS, pages 380-391. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-45803-7_32.
  2. Carla Binucci, Markus Chimani, Walter Didimo, Martin Gronemann, Karsten Klein, Jan Kratochvíl, Fabrizio Montecchiani, and Ioannis G. Tollis. Algorithms and characterizations for 2-layer fan-planarity: From caterpillar to stegosaurus. J. Graph Algorithms Appl., 21(1):81-102, 2017. URL: https://doi.org/10.7155/jgaa.00398.
  3. Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A survey on graph drawing beyond planarity. ACM Comput. Surv., 52(1):4:1-4:37, 2019. URL: https://doi.org/10.1145/3301281.
  4. V. Dujmović, M. Fellows, M. Kitching, G. Liotta, C. McCartin, N. Nishimura, P. Ragde, F. Rosamond, S. Whitesides, and D. Wood. On the parameterized complexity of layered graph drawing. Algorithmica, 52:267-292, 2008. URL: https://doi.org/10.1007/s00453-007-9151-1.
  5. S. Felsner, G. Liotta, and S. Wismath. Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl., 7(4):335-362, 2003. URL: https://doi.org/10.7155/jgaa.00075.
  6. L.S. Heath and A.L. Rosenberg. Laying out graphs using queues. SIAM Journal on Computing, 21(5):927-958, 1992. URL: https://doi.org/10.1137/0221055.
  7. Michael Kaufmann and Torsten Ueckerdt. The density of fan-planar graphs. CoRR, abs/1403.6184, 2014. URL: http://arxiv.org/abs/1403.6184.
  8. Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. An annotated bibliography on 1-planarity. Computer Science Review, 25:49-67, 2017. URL: https://doi.org/10.1016/j.cosrev.2017.06.002.
  9. M. Schaefer. Crossing Numbers of Graphs. CRC Press, 2018. Google Scholar
  10. M. Suderman. Pathwidth and layered drawings of trees. Intl. J. Comp. Geom. Appl., 14(3):203-225, 2004. URL: https://doi.org/10.1142/S0218195904001433.
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