29 Search Results for "Angelini, Patrizio"


Document
Enumeration of Intersection Graphs of x-Monotone Curves

Authors: Jacob Fox, János Pach, and Andrew Suk

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


Abstract
A curve in the plane is x-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct 2^Ω(n^{4/3}) families, each consisting of n labelled x-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most 2^O(n^{3/2-ε}), where ε > 0 is a suitable constant. Our proof uses an upper bound on the number of set systems of size m on a ground set of size n, with VC-dimension at most d. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

Cite as

Jacob Fox, János Pach, and Andrew Suk. Enumeration of Intersection Graphs of x-Monotone Curves. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 4:1-4:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{fox_et_al:LIPIcs.GD.2024.4,
  author =	{Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew},
  title =	{{Enumeration of Intersection Graphs of x-Monotone Curves}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{4:1--4:12},
  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.2024.4},
  URN =		{urn:nbn:de:0030-drops-212887},
  doi =		{10.4230/LIPIcs.GD.2024.4},
  annote =	{Keywords: Enumeration, intersection graphs, pseudo-segments, x-monotone}
}
Document
The Density Formula: One Lemma to Bound Them All

Authors: Michael Kaufmann, Boris Klemz, Kristin Knorr, Meghana M. Reddy, Felix Schröder, and Torsten Ueckerdt

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


Abstract
We introduce the Density Formula for (topological) drawings of graphs in the plane or on the sphere, which relates the number of edges, vertices, crossings, and sizes of cells in the drawing. We demonstrate its capability by providing several applications: we prove tight upper bounds on the edge density of various beyond-planar graph classes, including so-called k-planar graphs with k = 1,2, fan-crossing/fan-planar graphs, k-bend RAC-graphs with k = 0,1,2, quasiplanar graphs, and k^+-real face graphs. In some cases (1-bend and 2-bend RAC-graphs and fan-crossing/fan-planar graphs), we thereby obtain the first tight upper bounds on the edge density of the respective graph classes. In other cases, we give new streamlined and significantly shorter proofs for bounds that were already known in the literature. Thanks to the Density Formula, all of our proofs are mostly elementary counting and mostly circumvent the typical intricate case analysis found in earlier proofs. Further, in some cases (simple and non-homotopic quasiplanar graphs), our alternative proofs using the Density Formula lead to the first tight lower bound examples.

Cite as

Michael Kaufmann, Boris Klemz, Kristin Knorr, Meghana M. Reddy, Felix Schröder, and Torsten Ueckerdt. The Density Formula: One Lemma to Bound Them All. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{kaufmann_et_al:LIPIcs.GD.2024.7,
  author =	{Kaufmann, Michael and Klemz, Boris and Knorr, Kristin and M. Reddy, Meghana and Schr\"{o}der, Felix and Ueckerdt, Torsten},
  title =	{{The Density Formula: One Lemma to Bound Them All}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{7:1--7:17},
  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.2024.7},
  URN =		{urn:nbn:de:0030-drops-212913},
  doi =		{10.4230/LIPIcs.GD.2024.7},
  annote =	{Keywords: beyond-planar, density, fan-planar, fan-crossing, right-angle crossing, quasiplanar}
}
Document
The Parameterized Complexity Of Extending Stack Layouts

Authors: Thomas Depian, Simon D. Fink, Robert Ganian, and Martin Nöllenburg

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


Abstract
An 𝓁-page stack layout (also known as an 𝓁-page book embedding) of a graph is a linear order of the vertex set together with a partition of the edge set into 𝓁 stacks (or pages), such that the endpoints of no two edges on the same stack alternate. We study the problem of extending a given partial 𝓁-page stack layout into a complete one, which can be seen as a natural generalization of the classical NP-hard problem of computing a stack layout of an input graph from scratch. Given the inherent intractability of the problem, we focus on identifying tractable fragments through the refined lens of parameterized complexity analysis. Our results paint a detailed and surprisingly rich complexity-theoretic landscape of the problem which includes the identification of paraNP-hard, W[1]-hard and XP-tractable, as well as fixed-parameter tractable fragments of stack layout extension via a natural sequence of parameterizations.

Cite as

Thomas Depian, Simon D. Fink, Robert Ganian, and Martin Nöllenburg. The Parameterized Complexity Of Extending Stack Layouts. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{depian_et_al:LIPIcs.GD.2024.12,
  author =	{Depian, Thomas and Fink, Simon D. and Ganian, Robert and N\"{o}llenburg, Martin},
  title =	{{The Parameterized Complexity Of Extending Stack Layouts}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{12:1--12:17},
  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.2024.12},
  URN =		{urn:nbn:de:0030-drops-212960},
  doi =		{10.4230/LIPIcs.GD.2024.12},
  annote =	{Keywords: Stack Layout, Drawing Extension, Parameterized Complexity, Book Embedding}
}
Document
The Price of Upwardness

Authors: Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, and Alexander Wolff

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


Abstract
Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward k-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most k times for some integer k ≥ 1. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that upward-k-planarity testing is NP-complete already for k = 1 and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.

Cite as

Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, and Alexander Wolff. The Price of Upwardness. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{angelini_et_al:LIPIcs.GD.2024.13,
  author =	{Angelini, Patrizio and Biedl, Therese and Chimani, Markus and Cornelsen, Sabine and Da Lozzo, Giordano and Hong, Seok-Hee and Liotta, Giuseppe and Patrignani, Maurizio and Pupyrev, Sergey and Rutter, Ignaz and Wolff, Alexander},
  title =	{{The Price of Upwardness}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{13:1--13:20},
  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.2024.13},
  URN =		{urn:nbn:de:0030-drops-212977},
  doi =		{10.4230/LIPIcs.GD.2024.13},
  annote =	{Keywords: upward drawings, beyond planarity, upward k-planarity, upward outer-1-planarity}
}
Document
Bounding the Treewidth of Outer k-Planar Graphs via Triangulations

Authors: Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, and Alexander Wolff

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


Abstract
The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer k-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most k other edges. Wood and Telle [New York J. Math., 2007] showed that every outer k-planar graph has treewidth at most 3k + 11 using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer 1-planar graphs is at most 3, which is tight. In this paper, we improve the general upper bound to 1.5k + 2 and give a tight bound of 4 for k = 2. We also establish a lower bound: we show that, for every even k, there is an outer k-planar graph with treewidth k+2. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative. Our treewidth bound relies on a new and simple triangulation method for outer k-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of k + 2 for the separation number of outer k-planar graphs, improving an upper bound of 2k + 3 by Chaplick et al. [GD 2017]. We also consider outer min-k-planar graphs, a generalization of outer k-planar graphs, where we achieve smaller improvements.

Cite as

Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, and Alexander Wolff. Bounding the Treewidth of Outer k-Planar Graphs via Triangulations. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{firman_et_al:LIPIcs.GD.2024.14,
  author =	{Firman, Oksana and Gutowski, Grzegorz and Kryven, Myroslav and Okada, Yuto and Wolff, Alexander},
  title =	{{Bounding the Treewidth of Outer k-Planar Graphs via Triangulations}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{14:1--14:17},
  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.2024.14},
  URN =		{urn:nbn:de:0030-drops-212988},
  doi =		{10.4230/LIPIcs.GD.2024.14},
  annote =	{Keywords: treewidth, outerplanar graphs, outer k-planar graphs, outer min-k-planar graphs, cop number, separation number}
}
Document
Bundling-Aware Graph Drawing

Authors: Daniel Archambault, Giuseppe Liotta, Martin Nöllenburg, Tommaso Piselli, Alessandra Tappini, and Markus Wallinger

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


Abstract
Edge bundling algorithms significantly improve the visualization of dense graphs by reducing the clutter of many edges visible on screen by bundling them together. As such, bundling is often viewed as a post-processing step applied to a drawing, and the vast majority of edge bundling algorithms consider a graph and its drawing as input. Another way of thinking about edge bundling is to simultaneously optimize both the drawing and the bundling. In this paper, we investigate methods to simultaneously optimize a graph drawing and its bundling. We describe an algorithmic framework which consists of three main steps, namely Filter, Draw, and Bundle. We then propose two alternative implementations and experimentally compare them against the state-of-the-art approach and the simple idea of drawing and subsequently bundling the graph. The experiments confirm that bundled drawings created by our framework outperform previous approaches according to standard quality metrics for edge bundling.

Cite as

Daniel Archambault, Giuseppe Liotta, Martin Nöllenburg, Tommaso Piselli, Alessandra Tappini, and Markus Wallinger. Bundling-Aware Graph Drawing. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 15:1-15:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{archambault_et_al:LIPIcs.GD.2024.15,
  author =	{Archambault, Daniel and Liotta, Giuseppe and N\"{o}llenburg, Martin and Piselli, Tommaso and Tappini, Alessandra and Wallinger, Markus},
  title =	{{Bundling-Aware Graph Drawing}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{15:1--15:19},
  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.2024.15},
  URN =		{urn:nbn:de:0030-drops-212995},
  doi =		{10.4230/LIPIcs.GD.2024.15},
  annote =	{Keywords: Edge Bundling, Experimental Comparison, Graph Sparsification}
}
Document
Upward Pointset Embeddings of Planar st-Graphs

Authors: Carlos Alegría, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani

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


Abstract
We study upward pointset embeddings (UPSEs) of planar st-graphs. Let G be a planar st-graph and let S ⊂ ℝ² be a pointset with |S| = |V(G)|. An UPSE of G on S is an upward planar straight-line drawing of G that maps the vertices of G to the points of S. We consider both the problem of testing the existence of an UPSE of G on S (UPSE Testing) and the problem of enumerating all UPSEs of G on S. We prove that UPSE Testing is NP-complete even for st-graphs that consist of a set of directed st-paths sharing only s and t. On the other hand, for n-vertex planar st-graphs whose maximum st-cutset has size k, we prove that it is possible to solve UPSE Testing in 𝒪(n^{4k}) time with 𝒪(n^{3k}) space, and to enumerate all UPSEs of G on S with 𝒪(n) worst-case delay, using 𝒪(k n^{4k} log n) space, after 𝒪(k n^{4k} log n) set-up time. Moreover, for an n-vertex st-graph whose underlying graph is a cycle, we provide a necessary and sufficient condition for the existence of an UPSE on a given poinset, which can be tested in 𝒪(n log n) time. Related to this result, we give an algorithm that, for a set S of n points, enumerates all the non-crossing monotone Hamiltonian cycles on S with 𝒪(n) worst-case delay, using 𝒪(n²) space, after 𝒪(n²) set-up time.

Cite as

Carlos Alegría, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani. Upward Pointset Embeddings of Planar st-Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 24:1-24:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{alegria_et_al:LIPIcs.GD.2024.24,
  author =	{Alegr{\'\i}a, Carlos and Caroppo, Susanna and Da Lozzo, Giordano and D'Elia, Marco and Di Battista, Giuseppe and Frati, Fabrizio and Grosso, Fabrizio and Patrignani, Maurizio},
  title =	{{Upward Pointset Embeddings of Planar st-Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{24:1--24:18},
  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.2024.24},
  URN =		{urn:nbn:de:0030-drops-213082},
  doi =		{10.4230/LIPIcs.GD.2024.24},
  annote =	{Keywords: Upward pointset embeddings, planar st-graphs, st-cutset, non-crossing monotone Hamiltonian cycles, graph drawing enumeration}
}
Document
Parameterized Algorithms for Beyond-Planar Crossing Numbers

Authors: Miriam Münch and Ignaz Rutter

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


Abstract
Beyond-planar graph classes are usually defined via forbidden configurations or patterns in a drawing. In this paper, we formalize these concepts on a combinatorial level and show that, for any fixed family ℱ of crossing patterns, deciding whether a given graph G admits a drawing that avoids all patterns in F and that has at most c crossings is FPT w.r.t. c. In particular, we show that for any fixed k, deciding whether a graph is k-planar, k-quasi-planar, fan-crossing, fan-crossing-free or min-k-planar, respectively, is FPT with respect to the corresponding beyond-planar crossing number.

Cite as

Miriam Münch and Ignaz Rutter. Parameterized Algorithms for Beyond-Planar Crossing Numbers. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{munch_et_al:LIPIcs.GD.2024.25,
  author =	{M\"{u}nch, Miriam and Rutter, Ignaz},
  title =	{{Parameterized Algorithms for Beyond-Planar Crossing Numbers}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{25:1--25:16},
  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.2024.25},
  URN =		{urn:nbn:de:0030-drops-213096},
  doi =		{10.4230/LIPIcs.GD.2024.25},
  annote =	{Keywords: FPT, Beyond-planarity, Crossing-number, Crossing Patterns}
}
Document
On k-Planar Graphs Without Short Cycles

Authors: Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger

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


Abstract
We study the impact of forbidding short cycles to the edge density of k-planar graphs; a k-planar graph is one that can be drawn in the plane with at most k crossings per edge. Specifically, we consider three settings, according to which the forbidden substructures are 3-cycles, 4-cycles or both of them (i.e., girth ≥ 5). For all three settings and all k ∈ {1,2,3}, we present lower and upper bounds on the maximum number of edges in any k-planar graph on n vertices. Our bounds are of the form c n, for some explicit constant c that depends on k and on the setting. For general k ≥ 4 our bounds are of the form c√kn, for some explicit constant c. These results are obtained by leveraging different techniques, such as the discharging method, the recently introduced density formula for non-planar graphs, and new upper bounds for the crossing number of 2- and 3-planar graphs in combination with corresponding lower bounds based on the Crossing Lemma.

Cite as

Michael A. Bekos, Prosenjit Bose, Aaron Büngener, Vida Dujmović, Michael Hoffmann, Michael Kaufmann, Pat Morin, Saeed Odak, and Alexandra Weinberger. On k-Planar Graphs Without Short Cycles. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 27:1-27:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bekos_et_al:LIPIcs.GD.2024.27,
  author =	{Bekos, Michael A. and Bose, Prosenjit and B\"{u}ngener, Aaron and Dujmovi\'{c}, Vida and Hoffmann, Michael and Kaufmann, Michael and Morin, Pat and Odak, Saeed and Weinberger, Alexandra},
  title =	{{On k-Planar Graphs Without Short Cycles}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{27:1--27:17},
  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.2024.27},
  URN =		{urn:nbn:de:0030-drops-213117},
  doi =		{10.4230/LIPIcs.GD.2024.27},
  annote =	{Keywords: Beyond-planar Graphs, k-planar Graphs, Local Crossing Number, Crossing Number, Discharging Method, Crossing Lemma}
}
Document
On the Edge Density of Bipartite 3-Planar and Bipartite Gap-Planar Graphs

Authors: Aaron Büngener and Maximilian Pfister

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


Abstract
We show that if a bipartite graph G with n ≥ 3 vertices can be drawn in the plane such that (i) each edge is involved in at most three crossings per edge or (ii) each crossing is assigned to one of the two involved edges and each edge is assigned at most one crossing, then G has at most 4n-8 edges. In both cases, this bound is tight up to an additive constant as witnessed by lower-bound constructions. The former result can be used to improve the leading constant for the crossing lemma for bipartite graphs which in turn improves various results such as the biplanar crossing number or the maximum number of edges a bipartite k-planar graph can have.

Cite as

Aaron Büngener and Maximilian Pfister. On the Edge Density of Bipartite 3-Planar and Bipartite Gap-Planar Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 28:1-28:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bungener_et_al:LIPIcs.GD.2024.28,
  author =	{B\"{u}ngener, Aaron and Pfister, Maximilian},
  title =	{{On the Edge Density of Bipartite 3-Planar and Bipartite Gap-Planar Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{28:1--28:21},
  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.2024.28},
  URN =		{urn:nbn:de:0030-drops-213123},
  doi =		{10.4230/LIPIcs.GD.2024.28},
  annote =	{Keywords: Edge Density, Beyond Planarity, bipartite Graphs, Discharging Method}
}
Document
Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs

Authors: Aaron Büngener and Michael Kaufmann

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


Abstract
The classical Crossing Lemma by Ajtai et al. and Leighton from 1982 gave an important lower bound of cm³/n² for the number of crossings in any drawing of a given graph of n vertices and m edges. The original value was c = 1/100, which then has gradually been improved. Here, the bounds for the density of k-planar graphs played a central role. Our new insight is that for k = 2,3 the k-planar graphs have substantially fewer edges if specific local configurations that occur in drawings of k-planar graphs of maximum density are forbidden. Therefore, we are able to derive better bounds for the crossing number cr(G) of a given graph G. In particular, we achieve a bound of cr(G) ≥ 73/18m-305/18(n-2) for the range of 5n < m ≤ 6n, while our second bound cr(G) ≥ 5m - 407/18(n-2) is even stronger for larger m > 6n. For m > 6.79n, we finally apply the standard probabilistic proof from the BOOK and obtain an improved constant of c > 1/27.61 in the Crossing Lemma. Note that the previous constant was 1/29. Although this improvement is not too impressive, we consider our technique as an important new tool, which might be helpful in various other applications.

Cite as

Aaron Büngener and Michael Kaufmann. Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 29:1-29:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bungener_et_al:LIPIcs.GD.2024.29,
  author =	{B\"{u}ngener, Aaron and Kaufmann, Michael},
  title =	{{Improving the Crossing Lemma by Characterizing Dense 2-Planar and 3-Planar Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{29:1--29:22},
  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.2024.29},
  URN =		{urn:nbn:de:0030-drops-213136},
  doi =		{10.4230/LIPIcs.GD.2024.29},
  annote =	{Keywords: Crossing Lemma, k-planar graphs, discharging method}
}
Document
On the Complexity of Recognizing k^+-Real Face Graphs

Authors: Michael A. Bekos, Giuseppe Di Battista, Emilio Di Giacomo, Walter Didimo, Michael Kaufmann, and Fabrizio Montecchiani

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


Abstract
A nonplanar drawing Γ of a graph G divides the plane into topologically connected regions, called faces (or cells). The boundary of each face is formed by vertices, crossings, and edge segments. Given a positive integer k, we say that Γ is a k^+-real face drawing of G if the boundary of each face of Γ contains at least k vertices of G. The study of k^+-real face drawings started in a paper by Binucci et al. (WG 2023), where edge density bounds and relationships with other beyond-planar graph classes are proved. In this paper, we investigate the complexity of recognizing k^+-real face graphs, i.e., graphs that admit a k^+-real face drawing. We study both the general unconstrained scenario and the 2-layer scenario in which the graph is bipartite, the vertices of the two partition sets lie on two distinct horizontal layers, and the edges are straight-line segments. We give NP-completeness results for the unconstrained scenario and efficient recognition algorithms for the 2-layer setting.

Cite as

Michael A. Bekos, Giuseppe Di Battista, Emilio Di Giacomo, Walter Didimo, Michael Kaufmann, and Fabrizio Montecchiani. On the Complexity of Recognizing k^+-Real Face Graphs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 32:1-32:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{bekos_et_al:LIPIcs.GD.2024.32,
  author =	{Bekos, Michael A. and Di Battista, Giuseppe and Di Giacomo, Emilio and Didimo, Walter and Kaufmann, Michael and Montecchiani, Fabrizio},
  title =	{{On the Complexity of Recognizing k^+-Real Face Graphs}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{32:1--32:22},
  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.2024.32},
  URN =		{urn:nbn:de:0030-drops-213167},
  doi =		{10.4230/LIPIcs.GD.2024.32},
  annote =	{Keywords: Beyond planarity, k^+-real face drawings, 2-layer drawings, recognition algorithm, NP-hardness}
}
Document
On k-Plane Insertion into Plane Drawings

Authors: Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, and Ignaz Rutter

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


Abstract
We introduce the k-Plane Insertion into Plane drawing (k-PIP) problem: given a plane drawing of a planar graph G and a set F of edges, insert the edges in F into the drawing such that the resulting drawing is k-plane. In this paper, we show that the problem is NP-complete for every k ≥ 1, even when G is biconnected and the set F of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that k = 1 and G is a triangulation.

Cite as

Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, and Ignaz Rutter. On k-Plane Insertion into Plane Drawings. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 35:1-35:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{katheder_et_al:LIPIcs.GD.2024.35,
  author =	{Katheder, Julia and Kindermann, Philipp and Klute, Fabian and Parada, Irene and Rutter, Ignaz},
  title =	{{On k-Plane Insertion into Plane Drawings}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{35:1--35:11},
  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.2024.35},
  URN =		{urn:nbn:de:0030-drops-213190},
  doi =		{10.4230/LIPIcs.GD.2024.35},
  annote =	{Keywords: Graph drawing, edge insertion, k-planarity}
}
Document
Drawing Planar Graphs and 1-Planar Graphs Using Cubic Bézier Curves with Bounded Curvature

Authors: David Eppstein, Michael T. Goodrich, and Abraham M. Illickan

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


Abstract
We study algorithms for drawing planar graphs and 1-planar graphs using cubic Bézier curves with bounded curvature. We show that any n-vertex 1-planar graph has a 1-planar RAC drawing using a single cubic Bézier curve per edge, and this drawing can be computed in O(n) time given a combinatorial 1-planar drawing. We also show that any n-vertex planar graph G can be drawn in O(n) time with a single cubic Bézier curve per edge, in an O(n)× O(n) bounding box, such that the edges have Θ(1/degree(v)) angular resolution, for each v ∈ G, and O(√n) curvature.

Cite as

David Eppstein, Michael T. Goodrich, and Abraham M. Illickan. Drawing Planar Graphs and 1-Planar Graphs Using Cubic Bézier Curves with Bounded Curvature. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 39:1-39:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{eppstein_et_al:LIPIcs.GD.2024.39,
  author =	{Eppstein, David and Goodrich, Michael T. and Illickan, Abraham M.},
  title =	{{Drawing Planar Graphs and 1-Planar Graphs Using Cubic B\'{e}zier Curves with Bounded Curvature}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{39:1--39:17},
  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.2024.39},
  URN =		{urn:nbn:de:0030-drops-213237},
  doi =		{10.4230/LIPIcs.GD.2024.39},
  annote =	{Keywords: graph drawing, planar graphs, B\'{e}zier curves, and RAC drawings}
}
Document
Morphing Planar Graph Drawings via Orthogonal Box Drawings

Authors: Therese Biedl, Anna Lubiw, and Jack Spalding-Jamieson

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


Abstract
We give an algorithm to morph planar graph drawings that achieves small grid size at the expense of allowing a constant number of bends on each edge. The input is an n-vertex planar graph and two planar straight-line drawings of the graph on an O(n) × O(n) grid. The planarity-preserving morph is composed of O(n) linear morphs between successive pairs of drawings, each on an O(n) × O(n) grid with a constant number of bends per edge. The algorithm to compute the morph runs in O(n²) time on a word RAM model with standard arithmetic operations - in particular no square roots or cube roots are required. The first step of the algorithm is to morph each input drawing to a planar orthogonal box drawing where vertices are represented by boxes and each edge is drawn as a horizontal or vertical segment. The second step is to morph between planar orthogonal box drawings. This is done by extending known techniques for morphing planar orthogonal drawings with point vertices.

Cite as

Therese Biedl, Anna Lubiw, and Jack Spalding-Jamieson. Morphing Planar Graph Drawings via Orthogonal Box Drawings. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 40:1-40:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{biedl_et_al:LIPIcs.GD.2024.40,
  author =	{Biedl, Therese and Lubiw, Anna and Spalding-Jamieson, Jack},
  title =	{{Morphing Planar Graph Drawings via Orthogonal Box Drawings}},
  booktitle =	{32nd International Symposium on Graph Drawing and Network Visualization (GD 2024)},
  pages =	{40:1--40:16},
  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.2024.40},
  URN =		{urn:nbn:de:0030-drops-213242},
  doi =		{10.4230/LIPIcs.GD.2024.40},
  annote =	{Keywords: morphing, graph morphing, linear morph, planar graph drawing, orthogonal box drawing, orthogonal drawing, algorithm}
}
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