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DOI: 10.4230/LIPIcs.GD.2024.19

Weakly Leveled Planarity with Bounded Span

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

Published in: LIPIcs, Volume 320, 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)

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.

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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)

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@InProceedings{bekos_et_al:LIPIcs.GD.2024.19,
  author =	{Bekos, Michael A. and Da Lozzo, Giordano and Frati, Fabrizio and Gupta, Siddharth and Kindermann, Philipp and Liotta, Giuseppe and Rutter, Ignaz and Tollis, Ioannis G.},
  title =	{{Weakly Leveled Planarity with Bounded Span}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{19:1--19:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-343-0},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{320},
  editor =	{Felsner, Stefan and Klein, Karsten},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2024.19},
  URN =		{urn:nbn:de:0030-drops-213035},
  doi =		{10.4230/LIPIcs.GD.2024.19},
  annote =	{Keywords: Leveled planar graphs, edge span, graph drawing, edge-length ratio}
}

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DOI: 10.4230/LIPIcs.SEA.2023.2

Fast Reachability Using DAG Decomposition

Authors: Giorgos Kritikakis and Ioannis G. Tollis

Published in: LIPIcs, Volume 265, 21st International Symposium on Experimental Algorithms (SEA 2023)

Abstract

We present a fast and practical algorithm to compute the transitive closure (TC) of a directed graph. It is based on computing a reachability indexing scheme of a directed acyclic graph (DAG), G = (V, E). Given any path/chain decomposition of G we show how to compute in parameterized linear time such a reachability scheme that can answer reachability queries in constant time. The experimental results reveal that our method is significantly faster in practice than the theoretical bounds imply, indicating that path/chain decomposition algorithms can be applied to obtain fast and practical solutions to the transitive closure (TC) problem. Furthermore, we show that the number of non-transitive edges of a DAG G is ≤ width*|V| and that we can find a substantially large subset of the transitive edges of G in linear time using a path/chain decomposition. Our extensive experimental results show the interplay between these concepts in various models of DAGs.

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Giorgos Kritikakis and Ioannis G. Tollis. Fast Reachability Using DAG Decomposition. In 21st International Symposium on Experimental Algorithms (SEA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 265, pp. 2:1-2:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kritikakis_et_al:LIPIcs.SEA.2023.2,
  author =	{Kritikakis, Giorgos and Tollis, Ioannis G.},
  title =	{{Fast Reachability Using DAG Decomposition}},
  booktitle =	{21st International Symposium on Experimental Algorithms (SEA 2023)},
  pages =	{2:1--2:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-279-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{265},
  editor =	{Georgiadis, Loukas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SEA.2023.2},
  URN =		{urn:nbn:de:0030-drops-183526},
  doi =		{10.4230/LIPIcs.SEA.2023.2},
  annote =	{Keywords: graph algorithms, hierarchy, directed acyclic graphs (DAG), path/chain decomposition, transitive closure, transitive reduction, reachability, reachability indexing scheme}
}

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Documents authored by Tollis, Ioannis G.

Weakly Leveled Planarity with Bounded Span

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Fast Reachability Using DAG Decomposition

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