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Weakly Leveled Planarity with Bounded Span

Authors Michael A. Bekos , Giordano Da Lozzo , Fabrizio Frati , Siddharth Gupta , Philipp Kindermann , Giuseppe Liotta , Ignaz Rutter , Ioannis G. Tollis

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Author Details

Michael A. Bekos
  • University of Ioannina, Greece
Giordano Da Lozzo
  • Roma Tre University, Italy
Fabrizio Frati
  • Roma Tre University, Italy
Siddharth Gupta
  • BITS Pilani, K K Birla Goa Campus, India
Philipp Kindermann
  • Trier University, Germany
Giuseppe Liotta
  • Perugia University, Italy
Ignaz Rutter
  • University of Passau, Germany
Ioannis G. Tollis
  • University of Crete, Heraklion, Greece

Funding

Research by {Da Lozzo}, {Frati}, and {Liotta} was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND). Research by {Liotta} was also supported in part by MUR PON Proj. ARS01_00540. Research by Rutter was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306.

Acknowledgements

This work started at the Graph and Network Visualization Workshop 2023 (GNV '23), 25-30 June, Chania.

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Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Siddharth Gupta, Philipp Kindermann, Giuseppe Liotta, Ignaz Rutter, and Ioannis G. Tollis. Weakly Leveled Planarity with Bounded Span. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.19

Abstract

This paper studies planar drawings of graphs in which each vertex is represented as a point along a sequence of horizontal lines, called levels, and each edge is either a horizontal segment or a strictly y-monotone curve. A graph is s-span weakly leveled planar if it admits such a drawing where the edges have span at most s; the span of an edge is the number of levels it touches minus one. We investigate the problem of computing s-span weakly leveled planar drawings from both the computational and the combinatorial perspectives. We prove the problem to be para-NP-hard with respect to its natural parameter s and investigate its complexity with respect to widely used structural parameters. We show the existence of a polynomial-size kernel with respect to vertex cover number and prove that the problem is FPT when parameterized by treedepth. We also present upper and lower bounds on the span for various graph classes. Notably, we show that cycle trees, a family of 2-outerplanar graphs generalizing Halin graphs, are Θ(log n)-span weakly leveled planar and 4-span weakly leveled planar when 3-connected. As a byproduct of these combinatorial results, we obtain improved bounds on the edge-length ratio of the graph families under consideration.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • Leveled planar graphs
  • edge span
  • graph drawing
  • edge-length ratio

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