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Documents authored by Weinberger, Alexandra

Document
DOI: 10.4230/DagRep.15.5.64
Computational Geometry (Dagstuhl Seminar 25201)

Authors: Maarten Löffler, Eunjin Oh, Jeff M. Phillips, and Alexandra Weinberger

Published in: Dagstuhl Reports, Volume 15, Issue 5 (2025)

Abstract
This report documents the program and the outcomes of Dagstuhl Seminar 25201 "Computational Geometry". The seminar program spanned the days from 11th May to 16th May 2025, and 39 participants from various countries were on site. Recent advances in computational geometry were presented and discussed, and new challenges were identified, in particular in relation to the two themes "parameterized complexity" and "the interplay between theory and implementation". This report collects the abstracts of the talks and the open problems presented at the seminar, an excerpt from the panel discussion, and partial progress from the active working groups.
Cite as

Maarten Löffler, Eunjin Oh, Jeff M. Phillips, and Alexandra Weinberger. Computational Geometry (Dagstuhl Seminar 25201). In Dagstuhl Reports, Volume 15, Issue 5, pp. 64-95, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{loffler_et_al:DagRep.15.5.64,
  author =	{L\"{o}ffler, Maarten and Oh, Eunjin and Phillips, Jeff M. and Weinberger, Alexandra},
  title =	{{Computational Geometry (Dagstuhl Seminar 25201)}},
  pages =	{64--95},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2025},
  volume =	{15},
  number =	{5},
  editor =	{L\"{o}ffler, Maarten and Oh, Eunjin and Phillips, Jeff M. and Weinberger, Alexandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.15.5.64},
  URN =		{urn:nbn:de:0030-drops-252780},
  doi =		{10.4230/DagRep.15.5.64},
  annote =	{Keywords: algorithms, combinatorics, complexity, geometric computing, implementation}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.25
Characterizing and Recognizing Twistedness

Authors: Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
In a simple drawing of a graph, any two edges intersect in at most one point (either a common endpoint or a proper crossing). A simple drawing is generalized twisted if it fulfills certain rather specific constraints on how the edges are drawn. An abstract rotation system of a graph assigns to each vertex a cyclic order of its incident edges. A realizable rotation system is one that admits a simple drawing such that at each vertex, the edges emanate in that cyclic order, and a generalized twisted rotation system can be realized as a generalized twisted drawing. Generalized twisted drawings have initially been introduced to obtain improved bounds on the size of plane substructures in any simple drawing of K_n. They have since gained independent interest due to their surprising properties. However, the definition of generalized twisted drawings is very geometric and drawing-specific. In this paper, we develop characterizations of generalized twisted drawings that enable a purely combinatorial view on these drawings and lead to efficient recognition algorithms. Concretely, we show that for any n ≥ 7, an abstract rotation system of K_n is generalized twisted if and only if all subrotation systems induced by five vertices are generalized twisted. This implies a drawing-independent and concise characterization of generalized twistedness. Besides, the result yields a simple O(n⁵)-time algorithm to decide whether an abstract rotation system is generalized twisted and sheds new light on the structural features of simple drawings. We further develop a characterization via the rotations of a pair of vertices in a drawing, which we then use to derive an O(n²)-time algorithm to decide whether a realizable rotation system is generalized twisted.
Cite as

Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Characterizing and Recognizing Twistedness. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aichholzer_et_al:LIPIcs.GD.2025.25,
  author =	{Aichholzer, Oswin and Garc{\'\i}a, Alfredo and Tejel, Javier and Vogtenhuber, Birgit and Weinberger, Alexandra},
  title =	{{Characterizing and Recognizing Twistedness}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.25},
  URN =		{urn:nbn:de:0030-drops-250116},
  doi =		{10.4230/LIPIcs.GD.2025.25},
  annote =	{Keywords: generalized twisted drawings, simple drawings, rotation systems, recognition, combinatorial characterization, efficient algorithms}
}
Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2025.49
Defective Linear Layouts of Graphs (Poster Abstract)

Authors: Michael A. Bekos, Carla Binucci, Emilio Di Giacomo, Walter Didimo, Luca Grilli, Maria Eleni Pavlidi, Alessandra Tappini, and Alexandra Weinberger

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
A linear layout of a graph defines a total order of the vertices and partitions the edges into either stacks or queues, i.e., crossing-free and non-nested sets of edges along the order, respectively. In this work, we study defective linear layouts that allow forbidden patterns among edges of the same set. Our focus is on k-defective stack layouts and k-defective queue layouts, in which the conflict graph representing the forbidden patterns among the edges of each stack or queue has maximum degree at most k.
Cite as

Michael A. Bekos, Carla Binucci, Emilio Di Giacomo, Walter Didimo, Luca Grilli, Maria Eleni Pavlidi, Alessandra Tappini, and Alexandra Weinberger. Defective Linear Layouts of Graphs (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 49:1-49:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bekos_et_al:LIPIcs.GD.2025.49,
  author =	{Bekos, Michael A. and Binucci, Carla and Di Giacomo, Emilio and Didimo, Walter and Grilli, Luca and Pavlidi, Maria Eleni and Tappini, Alessandra and Weinberger, Alexandra},
  title =	{{Defective Linear Layouts of Graphs}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{49:1--49:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.49},
  URN =		{urn:nbn:de:0030-drops-250350},
  doi =		{10.4230/LIPIcs.GD.2025.49},
  annote =	{Keywords: Linear layouts, stack layouts, queue layouts, defective layouts}
}
Document
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2025.51
Graph Tiles (Poster Abstract)

Authors: Oswin Aichholzer, Robert Ganian, Phillip Keldenich, Maarten Löffler, Gert Meijer, Alexandra Weinberger, and Carola Wenk

Published in: LIPIcs, Volume 357, 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)

Abstract
We define a graph tile to be a unit square (or more generally, a polygon) on which a piece of a graph has been drawn/embedded; in particular, it may have vertices in its interior, edges connecting those vertices, or half-edges that extend to the boundary of the tile. In a graph tiling problem, we are given as input a set of graph tiles, with multiplicities, and the output is an arrangement of those tiles forming a graph of larger area. We focus on a simple tile set: unit square tiles with a central vertex and either a half-edge or no half-edge on each side. Up to symmetry this gives us six different types. We characterize which multiplicities are compatible for sets of at most three different tiles.
Cite as

Oswin Aichholzer, Robert Ganian, Phillip Keldenich, Maarten Löffler, Gert Meijer, Alexandra Weinberger, and Carola Wenk. Graph Tiles (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 51:1-51:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aichholzer_et_al:LIPIcs.GD.2025.51,
  author =	{Aichholzer, Oswin and Ganian, Robert and Keldenich, Phillip and L\"{o}ffler, Maarten and Meijer, Gert and Weinberger, Alexandra and Wenk, Carola},
  title =	{{Graph Tiles}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{51:1--51:5},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-403-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{357},
  editor =	{Dujmovi\'{c}, Vida and Montecchiani, Fabrizio},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.GD.2025.51},
  URN =		{urn:nbn:de:0030-drops-250371},
  doi =		{10.4230/LIPIcs.GD.2025.51},
  annote =	{Keywords: graph tiles}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.27
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
DOI: 10.4230/LIPIcs.SoCG.2023.6
Drawings of Complete Multipartite Graphs up to Triangle Flips

Authors: Oswin Aichholzer, Man-Kwun Chiu, Hung P. Hoang, Michael Hoffmann, Jan Kynčl, Yannic Maus, Birgit Vogtenhuber, and Alexandra Weinberger

Published in: LIPIcs, Volume 258, 39th International Symposium on Computational Geometry (SoCG 2023)

Abstract
For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan’s Theorem states that for any two simple drawings of the complete graph K_n with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on n vertices is bounded by O(n^{16}). The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the following sense: For the complete bipartite graph K_{m,n} minus two edges and K_{m,n} plus one edge for any m,n ≥ 4, as well as K_n minus a 4-cycle for any n ≥ 5, there exist two simple drawings with the same ERS that cannot be transformed into each other using triangle flips. So having the same ERS does not remain sufficient when removing or adding very few edges.
Cite as

Oswin Aichholzer, Man-Kwun Chiu, Hung P. Hoang, Michael Hoffmann, Jan Kynčl, Yannic Maus, Birgit Vogtenhuber, and Alexandra Weinberger. Drawings of Complete Multipartite Graphs up to Triangle Flips. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 6:1-6:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2023.6,
  author =	{Aichholzer, Oswin and Chiu, Man-Kwun and Hoang, Hung P. and Hoffmann, Michael and Kyn\v{c}l, Jan and Maus, Yannic and Vogtenhuber, Birgit and Weinberger, Alexandra},
  title =	{{Drawings of Complete Multipartite Graphs up to Triangle Flips}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{6:1--6:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-273-0},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{258},
  editor =	{Chambers, Erin W. and Gudmundsson, Joachim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2023.6},
  URN =		{urn:nbn:de:0030-drops-178563},
  doi =		{10.4230/LIPIcs.SoCG.2023.6},
  annote =	{Keywords: Simple drawings, simple topological graphs, complete graphs, multipartite graphs, k-partite graphs, bipartite graphs, Gioan’s Theorem, triangle flips, Reidemeister moves}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.5
Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs

Authors: Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized twisted drawings. A simple drawing is generalized twisted if there is a point O such that every ray emanating from O crosses every edge of the drawing at most once and there is a ray emanating from O which crosses every edge exactly once. Via this new class of simple drawings, we show that every simple drawing of the complete graph with n vertices contains Ω(n^{1/2}) pairwise disjoint edges and a plane path of length Ω((log n)/(log log n)). Both results improve over previously known best lower bounds. On the way we show several structural results about and properties of generalized twisted drawings. We further present different characterizations of generalized twisted drawings, which might be of independent interest.
Cite as

Oswin Aichholzer, Alfredo García, Javier Tejel, Birgit Vogtenhuber, and Alexandra Weinberger. Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 5:1-5:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{aichholzer_et_al:LIPIcs.SoCG.2022.5,
  author =	{Aichholzer, Oswin and Garc{\'\i}a, Alfredo and Tejel, Javier and Vogtenhuber, Birgit and Weinberger, Alexandra},
  title =	{{Twisted Ways to Find Plane Structures in Simple Drawings of Complete Graphs}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{5:1--5:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.5},
  URN =		{urn:nbn:de:0030-drops-160136},
  doi =		{10.4230/LIPIcs.SoCG.2022.5},
  annote =	{Keywords: Simple drawings, simple topological graphs, disjoint edges, plane matching, plane path}
}
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