24 Search Results for "Suk, Andrew"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.63
Unavoidable Patterns and Plane Paths in Dense Topological Graphs

Authors: Balázs Keszegh, Andrew Suk, Gábor Tardos, and Ji Zeng

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Let C_{s,t} be the complete bipartite geometric graph, with s and t vertices on two distinct parallel lines respectively, and all s t straight-line edges drawn between them. In this paper, we show that every complete bipartite simple topological graph, with parts of size 2(k-1)⁴ + 1 and 2^{k^{5k}}, contains a topological subgraph weakly isomorphic to C_{k,k}. As a corollary, every n-vertex simple topological graph not containing a plane path of length k has at most O_k(n^{2 - 8/k⁴}) edges. When k = 3, we obtain a stronger bound by showing that every n-vertex simple topological graph not containing a plane path of length 3 has at most O(n^{4/3}) edges. We also prove that x-monotone simple topological graphs not containing a plane path of length 3 have at most a linear number of edges.
Cite as

Balázs Keszegh, Andrew Suk, Gábor Tardos, and Ji Zeng. Unavoidable Patterns and Plane Paths in Dense Topological Graphs. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 63:1-63:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{keszegh_et_al:LIPIcs.SoCG.2026.63,
  author =	{Keszegh, Bal\'{a}zs and Suk, Andrew and Tardos, G\'{a}bor and Zeng, Ji},
  title =	{{Unavoidable Patterns and Plane Paths in Dense Topological Graphs}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{63:1--63:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.63},
  URN =		{urn:nbn:de:0030-drops-258706},
  doi =		{10.4230/LIPIcs.SoCG.2026.63},
  annote =	{Keywords: graph drawing, topological graph, bipartite geometric graph, forbidden subgraph, extremal graph, thrackle}
}
Document
Research
DOI: 10.4230/TGDK.3.3.4
Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web

Authors: Florian Ruosch, Cristina Sarasua, and Abraham Bernstein

Published in: TGDK, Volume 3, Issue 3 (2025). Transactions on Graph Data and Knowledge, Volume 3, Issue 3

Abstract
In Argument Mining, predicting argumentative relations between texts (or spans) remains one of the most challenging aspects, even more so in the cross-document setting. This paper makes three key contributions to advance research in this domain. We first extend an existing dataset, the Sci-Arg corpus, by annotating it with explicit inter-document argumentative relations, thereby allowing arguments to be distributed over several documents forming an Argument Web; these new annotations are published using Semantic Web technologies (RDF, OWL). Second, we explore and evaluate three automated approaches for predicting these inter-document argumentative relations, establishing critical baselines on the new dataset. We find that a simple classifier based on discourse indicators with access to context outperforms neural methods. Third, we conduct a comparative analysis of these approaches for both intra- and inter-document settings, identifying statistically significant differences in results that indicate the necessity of distinguishing between these two scenarios. Our findings highlight significant challenges in this complex domain and open crucial avenues for future research on the Argument Web of Science, particularly for those interested in leveraging Semantic Web technologies and knowledge graphs to understand scholarly discourse. With this, we provide the first stepping stones in the form of a benchmark dataset, three baseline methods, and an initial analysis for a systematic exploration of this field relevant to the Web of Data and Science.
Cite as

Florian Ruosch, Cristina Sarasua, and Abraham Bernstein. Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web. In Transactions on Graph Data and Knowledge (TGDK), Volume 3, Issue 3, pp. 4:1-4:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@Article{ruosch_et_al:TGDK.3.3.4,
  author =	{Ruosch, Florian and Sarasua, Cristina and Bernstein, Abraham},
  title =	{{Mining Inter-Document Argument Structures in Scientific Papers for an Argument Web}},
  journal =	{Transactions on Graph Data and Knowledge},
  pages =	{4:1--4:33},
  ISSN =	{2942-7517},
  year =	{2025},
  volume =	{3},
  number =	{3},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/TGDK.3.3.4},
  URN =		{urn:nbn:de:0030-drops-252159},
  doi =		{10.4230/TGDK.3.3.4},
  annote =	{Keywords: Argument Mining, Large Language Models, Knowledge Graphs, Link Prediction}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.21
On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers

Authors: Parinya Chalermsook, Ly Orgo, and Minoo Zarsav

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

Abstract
This paper considers the Zarankiewicz problem in bipartite graphs with low-dimensional geometric representation (i.e., low Ferrers dimension). Let Z(n;k) be the maximum number of edges in a bipartite graph with n nodes and is free of a k-by-k biclique. Note that Z(n;k) ∈ Ω(nk) for all "natural" graph classes. Our first result reveals a separation between bipartite graphs of Ferrers dimension three and four: while we show that Z(n;k) ≤ 9n(k-1) for graphs of Ferrers dimension three, Z(n;k) ∈ Ω(n k ⋅ (log n)/(log log n)) for Ferrers dimension four graphs (Chan & Har-Peled, 2023) (Chazelle, 1990). To complement this, we derive a tight upper bound of 2n(k-1) for chordal bipartite graphs and 54n(k-1) for grid intersection graphs (GIG), a prominent graph class residing in four Ferrers dimensions and capturing planar bipartite graphs as well as bipartite intersection graphs of rectangles. Previously, the best-known bound for GIG was Z(n;k) ∈ O(2^{O(k)} n), implied by the results of Fox & Pach (2006) and Mustafa & Pach (2016). Our results advance and offer new insights into the interplay between Ferrers dimensions and extremal combinatorics.
Cite as

Parinya Chalermsook, Ly Orgo, and Minoo Zarsav. On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 21:1-21:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chalermsook_et_al:LIPIcs.GD.2025.21,
  author =	{Chalermsook, Parinya and Orgo, Ly and Zarsav, Minoo},
  title =	{{On Geometric Bipartite Graphs with Asymptotically Smallest Zarankiewicz Numbers}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{21:1--21:24},
  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.2025.21},
  URN =		{urn:nbn:de:0030-drops-250074},
  doi =		{10.4230/LIPIcs.GD.2025.21},
  annote =	{Keywords: Bipartite graph classes, extremal graph theory, geometric intersection graphs, Zarankiewicz problem, bicliques}
}
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
DOI: 10.4230/LIPIcs.GD.2025.10
The Bend Number of Cocomparability Graphs

Authors: Todor Antić, Vit Jelínek, Martin Pergel, Felix Schröder, Peter Stumpf, and Pavel Valtr

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

Abstract
We introduce a new complexity measure for cocomparability graphs of posets or in other words, intersection graphs of piecewise linear functions, the bend number. We prove that cocomparability graphs of bounded bend number are not too plentiful and give two hierarchies of classes of cocomparability graphs, depending on whether the piecewise linear functions are restricted to slopes of ±1 (diagonal case) or not (general case). These hierarchies give a gradation between permutation graphs and cocomparability graphs.
Cite as

Todor Antić, Vit Jelínek, Martin Pergel, Felix Schröder, Peter Stumpf, and Pavel Valtr. The Bend Number of Cocomparability Graphs. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 10:1-10:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{antic_et_al:LIPIcs.GD.2025.10,
  author =	{Anti\'{c}, Todor and Jel{\'\i}nek, Vit and Pergel, Martin and Schr\"{o}der, Felix and Stumpf, Peter and Valtr, Pavel},
  title =	{{The Bend Number of Cocomparability Graphs}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{10:1--10:23},
  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.2025.10},
  URN =		{urn:nbn:de:0030-drops-249963},
  doi =		{10.4230/LIPIcs.GD.2025.10},
  annote =	{Keywords: Intersection Graphs, Bend Number, Piecewise Linear Functions, Graph Class Hierarchy, Cocomparability Graphs, Permutation Graphs, Poset Dimension}
}
Document
DOI: 10.4230/LIPIcs.GD.2025.31
From Local Pair-Crossing Number to Local Crossing Number

Authors: Jacob Fox, János Pach, and Andrew Suk

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

Abstract
We prove that if a graph can be drawn in the plane such that each edge crosses at most k other edges, then it can be redrawn so that each edge participates in at most k³+O(k²) crossings. This improves the previous exponential bound that follows from a result of Schaefer and Štefankovič and answers a question of Ackerman and Schaefer.
Cite as

Jacob Fox, János Pach, and Andrew Suk. From Local Pair-Crossing Number to Local Crossing Number. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 31:1-31:6, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fox_et_al:LIPIcs.GD.2025.31,
  author =	{Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew},
  title =	{{From Local Pair-Crossing Number to Local Crossing Number}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{31:1--31:6},
  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.2025.31},
  URN =		{urn:nbn:de:0030-drops-250170},
  doi =		{10.4230/LIPIcs.GD.2025.31},
  annote =	{Keywords: Crossing numbers, pair crossing numbers}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.67
Compact Representation of Semilinear and Terrain-Like Graphs

Authors: Jean Cardinal and Yelena Yuditsky

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We consider the existence and construction of biclique covers of graphs, consisting of coverings of their edge sets by complete bipartite graphs. The size of such a cover is the sum of the sizes of the bicliques. Small-size biclique covers of graphs are ubiquitous in computational geometry, and have been shown to be useful compact representations of graphs. We give a brief survey of classical and recent results on biclique covers and their applications, and give new families of graphs having biclique covers of near-linear size. In particular, we show that semilinear graphs, whose edges are defined by linear relations in bounded dimensional space, always have biclique covers of size O(npolylog n). This generalizes many previously known results on special classes of graphs including interval graphs, permutation graphs, and graphs of bounded boxicity, but also new classes such as intersection graphs of L-shapes in the plane. It also directly implies the bounds for Zarankiewicz’s problem derived by Basit, Chernikov, Starchenko, Tao, and Tran (Forum Math. Sigma, 2021). We also consider capped graphs, also known as terrain-like graphs, defined as ordered graphs forbidding a certain ordered pattern on four vertices. Terrain-like graphs contain the induced subgraphs of terrain visibility graphs. We give an elementary proof that these graphs admit biclique partitions of size O(nlog³ n). This provides a simple combinatorial analogue of a classical result from Agarwal, Alon, Aronov, and Suri on polygon visibility graphs (Discrete Comput. Geom. 1994). Finally, we prove that there exists families of unit disk graphs on n vertices that do not admit biclique coverings of size o(n^{4/3}), showing that we are unlikely to improve on Szemerédi-Trotter type incidence bounds for higher-degree semialgebraic graphs.
Cite as

Jean Cardinal and Yelena Yuditsky. Compact Representation of Semilinear and Terrain-Like Graphs. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 67:1-67:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cardinal_et_al:LIPIcs.ESA.2025.67,
  author =	{Cardinal, Jean and Yuditsky, Yelena},
  title =	{{Compact Representation of Semilinear and Terrain-Like Graphs}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{67:1--67:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.67},
  URN =		{urn:nbn:de:0030-drops-245359},
  doi =		{10.4230/LIPIcs.ESA.2025.67},
  annote =	{Keywords: Biclique covers, intersection graphs, visibility graphs, Zarankiewicz’s problem}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.36
Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs

Authors: James Davies, Agelos Georgakopoulos, Meike Hatzel, and Rose McCarty

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
In this paper, we consider the class 𝒞^d of sphere intersection graphs in R^d for d ≥ 2. We show that for each integer t, the class of all graphs in 𝒞^d that exclude K_{t,t} as a subgraph has strongly sublinear separators. We also prove that 𝒞^d has asymptotic dimension at most 2d+2.
Cite as

James Davies, Agelos Georgakopoulos, Meike Hatzel, and Rose McCarty. Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 36:1-36:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{davies_et_al:LIPIcs.SoCG.2025.36,
  author =	{Davies, James and Georgakopoulos, Agelos and Hatzel, Meike and McCarty, Rose},
  title =	{{Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{36:1--36:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.36},
  URN =		{urn:nbn:de:0030-drops-231881},
  doi =		{10.4230/LIPIcs.SoCG.2025.36},
  annote =	{Keywords: Intersection graphs, strongly sublinear separators, asymptotic dimension}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.19
Polychromatic Coloring of Tuples in Hypergraphs

Authors: Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
A hypergraph H consists of a set V of vertices and a set E of hyperedges that are subsets of V. A t-tuple of H is a subset of t vertices of V. A t-tuple k-coloring of H is a mapping of its t-tuples into k colors. A coloring is called (t,k,f)-polychromatic if each hyperedge of E that has at least f vertices contains tuples of all the k colors. Let f_H(t,k) be the minimum f such that H has a (t,k,f)-polychromatic coloring. For a family of hypergraphs ℋ let f_H(t,k) be the maximum f_H(t,k) over all hypergraphs H in H. Determining f_H(t,k) has been an active research direction in recent years. This is challenging even for t = 1. We present several new results in this direction for t ≥ 2. - Let H be the family of hypergraphs H that is obtained by taking any set P of points in ℝ², setting V: = P and E: = {d ∩ P: d is a disk in ℝ²}. We prove that f_ H(2,k) ≤ 3.7^k, that is, the pairs of points (2-tuples) can be k-colored such that any disk containing at least 3.7^k points has pairs of all colors. We generalize this result to points and balls in higher dimensions. - For the family H of hypergraphs that are defined by grid vertices and axis-parallel rectangles in the plane, we show that f_H(2,k) ≤ √{ck ln k} for some constant c. We then generalize this to higher dimensions, to other shapes, and to tuples of larger size. - For the family H of shrinkable hypergraphs of VC-dimension at most d we prove that f_ H(d+1,k) ≤ c^k for some constant c = c(d). Towards this bound, we obtain a result of independent interest: Every hypergraph with n vertices and with VC-dimension at most d has a (d+1)-tuple T of depth at least n/c, i.e., any hyperedge that contains T also contains n/c other vertices. - For the relationship between t-tuple coloring and vertex coloring in any hypergraph H we establish the inequality 1/e⋅ tk^{1/t} ≤ f_H(t,k) ≤ f_H(1,tk^{1/t}). For the special case of k = 2, referred to as the bichromatic coloring, we prove that t+1 ≤ f_H(t,2) ≤ max{f_H(1,2), t+1}; this improves upon the previous best known upper bound. - We study the relationship between tuple coloring and epsilon nets. In particular we show that if f_H(1,k) = O(k) for a hypergraph H with n vertices, then for any 0 < ε < 1 the t-tuples of H can be partitioned into Ω((εn/t)^t) ε-t-nets. This bound is tight when t is a constant.
Cite as

Ahmad Biniaz, Jean-Lou De Carufel, Anil Maheshwari, Michiel Smid, Shakhar Smorodinsky, and Miloš Stojaković. Polychromatic Coloring of Tuples in Hypergraphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 19:1-19:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{biniaz_et_al:LIPIcs.SoCG.2025.19,
  author =	{Biniaz, Ahmad and De Carufel, Jean-Lou and Maheshwari, Anil and Smid, Michiel and Smorodinsky, Shakhar and Stojakovi\'{c}, Milo\v{s}},
  title =	{{Polychromatic Coloring of Tuples in Hypergraphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{19:1--19:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.19},
  URN =		{urn:nbn:de:0030-drops-231718},
  doi =		{10.4230/LIPIcs.SoCG.2025.19},
  annote =	{Keywords: Hypergraph Coloring, Polychromatic Coloring, Geometric Hypergraphs, Cover Decomposable Hypergraphs, Epsilon Nets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.17
Sparse Bounded Hop-Spanners for Geometric Intersection Graphs

Authors: Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
We present new results on 2- and 3-hop spanners for geometric intersection graphs. These include improved upper and lower bounds for 2- and 3-hop spanners for many geometric intersection graphs in ℝ^d. For example, we show that the intersection graph of n balls in ℝ^d admits a 2-hop spanner of size O^*(n^{3/2 - 1/(2(2⌊d/2⌋ + 1))}) and the intersection graph of n fat axis-parallel boxes in ℝ^d admits a 2-hop spanner of size O(n log^{d+1}n). Furthermore, we show that the intersection graph of general semi-algebraic objects in ℝ^d admits a 3-hop spanner of size O^*(n^{3/2 - 1/(2(2D-1))}), where D is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in ℝ³), we provide a lower bound of Ω(n^{4/3}). For 3-hop and axis-parallel boxes in ℝ^d, we provide the upper bound O(n log ^{d-1}n) and lower bound Ω(n ({log n}/{log log n})^{d-2}).
Cite as

Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth. Sparse Bounded Hop-Spanners for Geometric Intersection Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhore_et_al:LIPIcs.SoCG.2025.17,
  author =	{Bhore, Sujoy and Chan, Timothy M. and Huang, Zhengcheng and Smorodinsky, Shakhar and T\'{o}th, Csaba D.},
  title =	{{Sparse Bounded Hop-Spanners for Geometric Intersection Graphs}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{17:1--17:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.17},
  URN =		{urn:nbn:de:0030-drops-231698},
  doi =		{10.4230/LIPIcs.SoCG.2025.17},
  annote =	{Keywords: Geometric Spanners, Geometric Intersection Graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.50
Immersions and Albertson’s Conjecture

Authors: Jacob Fox, János Pach, and Andrew Suk

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
A graph is said to contain K_k (a clique of size k) as a weak immersion if it has k vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger’s conjecture: Every graph of chromatic number k contains K_k as a weak immersion. We prove this conjecture for graphs with at most 1.4(k-1) vertices. As an application, we make some progress on Albertson’s conjecture on crossing numbers of graphs, according to which every graph G with chromatic number k satisfies cr(G) ≥ cr(K_k). In particular, we show that the conjecture is true for all graphs of chromatic number k, provided that they have at most 1.4(k-1) vertices and k is sufficiently large.
Cite as

Jacob Fox, János Pach, and Andrew Suk. Immersions and Albertson’s Conjecture. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 50:1-50:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2025.50,
  author =	{Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew},
  title =	{{Immersions and Albertson’s Conjecture}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{50:1--50:10},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.50},
  URN =		{urn:nbn:de:0030-drops-232022},
  doi =		{10.4230/LIPIcs.SoCG.2025.50},
  annote =	{Keywords: Immersions, crossing numbers, chromatic number}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.76
A Note on the No-(d+2)-On-a-Sphere Problem

Authors: Andrew Suk and Ethan Patrick White

Published in: LIPIcs, Volume 332, 41st International Symposium on Computational Geometry (SoCG 2025)

Abstract
For fixed d ≥ 3, we construct subsets of the d-dimensional lattice cube [n]^d of size n^{3/(d + 1) - o(1)} with no d+2 points on a sphere or a hyperplane. This improves the previously best known bound of Ω(n^{1/(d-1)}) due to Thiele from 1995.
Cite as

Andrew Suk and Ethan Patrick White. A Note on the No-(d+2)-On-a-Sphere Problem. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 76:1-76:8, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{suk_et_al:LIPIcs.SoCG.2025.76,
  author =	{Suk, Andrew and White, Ethan Patrick},
  title =	{{A Note on the No-(d+2)-On-a-Sphere Problem}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{76:1--76:8},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-370-6},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{332},
  editor =	{Aichholzer, Oswin and Wang, Haitao},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2025.76},
  URN =		{urn:nbn:de:0030-drops-232287},
  doi =		{10.4230/LIPIcs.SoCG.2025.76},
  annote =	{Keywords: General position, no-four-on-a-cirle, d-dimensional lattice cube}
}
Document
DOI: 10.4230/LIPIcs.GD.2024.4
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
DOI: 10.4230/LIPIcs.SoCG.2024.59
A Structure Theorem for Pseudo-Segments and Its Applications

Authors: Jacob Fox, János Pach, and Andrew Suk

Published in: LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024)

Abstract
We prove a far-reaching strengthening of Szemerédi’s regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such graphs can be partitioned into a bounded number of parts of roughly the same size such that almost all of the bipartite graphs between pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every n-vertex simple topological graph with no k pairwise disjoint edges has at most n(log n)^O(log k) edges.
Cite as

Jacob Fox, János Pach, and Andrew Suk. A Structure Theorem for Pseudo-Segments and Its Applications. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fox_et_al:LIPIcs.SoCG.2024.59,
  author =	{Fox, Jacob and Pach, J\'{a}nos and Suk, Andrew},
  title =	{{A Structure Theorem for Pseudo-Segments and Its Applications}},
  booktitle =	{40th International Symposium on Computational Geometry (SoCG 2024)},
  pages =	{59:1--59:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-316-4},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{293},
  editor =	{Mulzer, Wolfgang and Phillips, Jeff M.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.59},
  URN =		{urn:nbn:de:0030-drops-200040},
  doi =		{10.4230/LIPIcs.SoCG.2024.59},
  annote =	{Keywords: Regularity lemma, pseudo-segments, intersection graphs}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2023.41
Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems

Authors: Alfredo Hubard and Andrew Suk

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

Abstract
Given a complete simple topological graph G, a k-face generated by G is the open bounded region enclosed by the edges of a non-self-intersecting k-cycle in G. Interestingly, there are complete simple topological graphs with the property that every odd face it generates contains the origin. In this paper, we show that every complete n-vertex simple topological graph generates at least Ω(n^{1/3}) pairwise disjoint 4-faces. As an immediate corollary, every complete simple topological graph on n vertices drawn in the unit square generates a 4-face with area at most O(n^{-1/3}). Finally, we investigate a ℤ₂ variant of Heilbronn’s triangle problem for not necessarily simple complete topological graphs.
Cite as

Alfredo Hubard and Andrew Suk. Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 41:1-41:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hubard_et_al:LIPIcs.SoCG.2023.41,
  author =	{Hubard, Alfredo and Suk, Andrew},
  title =	{{Disjoint Faces in Drawings of the Complete Graph and Topological Heilbronn Problems}},
  booktitle =	{39th International Symposium on Computational Geometry (SoCG 2023)},
  pages =	{41:1--41:15},
  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.2023.41},
  URN =		{urn:nbn:de:0030-drops-178917},
  doi =		{10.4230/LIPIcs.SoCG.2023.41},
  annote =	{Keywords: Disjoint faces, simple topological graphs, topological Heilbronn problems}
}
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