17 Search Results for "Kynčl, Jan"

Document
DOI: 10.4230/LIPIcs.WG.2026.15
High Beer Index Implies Big Hollow Triangles

Authors: Arun Kumar Das, Vít Jelínek, Jan Kynčl, Martin Pergel, Felix Schröder, Peter Stumpf, and Pavel Valtr

Published in: LIPIcs, Volume 376, 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)

Abstract
The visibility graph of a set S ⊆ ℝ² is the graph whose vertices are the points of S, with two points x,y connected by an edge if and only if they see each other in S, that is, if the segment xy is contained in S. The edge density of this graph is known as the Beer index of S. Previously, it has been shown that a simply connected set S ⊆ ℝ² of unit Lebesgue measure with Beer index β > 0 contains a convex subset of measure Ω(β); in particular, for visibility graphs of simply connected sets, a positive edge density β > 0 implies the existence of a clique containing an Ω(β)-fraction of all vertices. The simple-connectivity assumption cannot be omitted, as there are non-simply-connected sets with Beer index 1 and no convex subset of positive measure. Nevertheless, in this paper, we extend the above result to non-simply-connected sets, by showing that a visibility graph with large edge density contains a triangle with large convex hull. More precisely, we show that a set S ⊆ ℝ² of unit Lebesgue measure with Beer index β > 0 contains three pairwise visible points whose convex hull has measure Ω(β⁹). If in addition S is an open domain with K holes, then S contains three pairwise visible points with convex hull of measure Ω(β/K) as well as a convex subset of measure Ω(β/K²).
Cite as

Arun Kumar Das, Vít Jelínek, Jan Kynčl, Martin Pergel, Felix Schröder, Peter Stumpf, and Pavel Valtr. High Beer Index Implies Big Hollow Triangles. In 52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 376, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{das_et_al:LIPIcs.WG.2026.15,
  author =	{Das, Arun Kumar and Jel{\'\i}nek, V{\'\i}t and Kyn\v{c}l, Jan and Pergel, Martin and Schr\"{o}der, Felix and Stumpf, Peter and Valtr, Pavel},
  title =	{{High Beer Index Implies Big Hollow Triangles}},
  booktitle =	{52nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2026)},
  pages =	{15:1--15:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-430-7},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{376},
  editor =	{Goedgebeur, Jan and Rz\k{a}\.{z}ewski, Pawe{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WG.2026.15},
  URN =		{urn:nbn:de:0030-drops-261813},
  doi =		{10.4230/LIPIcs.WG.2026.15},
  annote =	{Keywords: convexity, Beer index, visibility graph}
}
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
DOI: 10.4230/LIPIcs.SoCG.2026.9
Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions

Authors: Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc

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

Abstract
A theorem of Matoušek asserts that for any k ≥ 2, any set system whose shatter function is o(n^k) enjoys a fractional Helly theorem of order k: in the k-wise intersection hypergraph, positive density implies a linear-size clique. Kalai and Meshulam conjectured a generalization of that phenomenon to homological shatter functions. It was verified for set systems with bounded homological shatter functions and whose ground set has a forbidden homological minor (which includes ℝ^d by a homological analogue of the van Kampen-Flores theorem). We present two contributions to this line of research: - We study homological minors in certain manifolds (possibly with boundary), for which we prove analogues of the van Kampen-Flores theorem and of the Hanani-Tutte theorem. - We introduce graded analogues of the Radon and Helly numbers of set systems and relate their growth rate to the original parameters. This allows to extend the verification of the Kalai-Meshulam conjecture to sufficiently slowly growing homological shatter functions.
Cite as

Sergey Avvakumov, Marguerite Bin, and Xavier Goaoc. Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{avvakumov_et_al:LIPIcs.SoCG.2026.9,
  author =	{Avvakumov, Sergey and Bin, Marguerite and Goaoc, Xavier},
  title =	{{Intersection Patterns of Set Systems on Manifolds with Slowly Growing Homological Shatter Functions}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{9:1--9:16},
  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.2026.9},
  URN =		{urn:nbn:de:0030-drops-258152},
  doi =		{10.4230/LIPIcs.SoCG.2026.9},
  annote =	{Keywords: Fractional Helly theorem, homological minor, combinatorial convexity}
}
Document
DOI: 10.4230/LIPIcs.FUN.2026.8
Directed Grabbing Games or How to Politely Grab the Maximum Number of Olives in a Reception

Authors: Jean-Claude Bermond, Michel Cosnard, Frédéric Havet, Takako Kodate, and Stéphane Pérennes

Published in: LIPIcs, Volume 366, 13th International Conference on Fun with Algorithms (FUN 2026)

Abstract
We introduce and study the directed grabbing game, a directed variation of the graph grabbing game played by two players Alice and Bob on a weighted acyclic digraph D. Alice plays first and then they play alternately. At a given odd (resp. even) move, Alice (resp. Bob) chooses a sink, that is a vertex of out-degree 0, grabs the weight (olives) on it and then removes the vertex. The aim of each player is to grab a maximum weight, i.e. a maximum number of olives. This game is inspired by the behaviour that guests are expected to adopt during a reception or cocktail party. We first consider the case where hors d'oeuvre are arranged on slightly spaced parallel lines, such that politeness allows one to take the first hors d'oeuvre from each line. This corresponds to the directed grabbing game on a union of disjoint directed paths. We give an algorithm that, given a weighted digraph D of order n which is the union of q disjoint directed paths, computes an optimal play in O(nlog q) time. Then we consider the "pissaladière case" where the digraph D is a directed (p× q)-grid. We show that, depending on the parity of pq, one player, called Content, has a strategic advantage. Specifically, Content is Alice when pq is odd and Bob when pq is even. We present some strategies that enable Content to remove some large sets of vertices (of order pq/2) in directed grids. We then derive that Content can remove any given vertex that is not in the border of the grid. Finally, in the case where each vertex contains either zero or one olive, we prove that Content can secure the grabbing of around one third of the olives.
Cite as

Jean-Claude Bermond, Michel Cosnard, Frédéric Havet, Takako Kodate, and Stéphane Pérennes. Directed Grabbing Games or How to Politely Grab the Maximum Number of Olives in a Reception. In 13th International Conference on Fun with Algorithms (FUN 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 366, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bermond_et_al:LIPIcs.FUN.2026.8,
  author =	{Bermond, Jean-Claude and Cosnard, Michel and Havet, Fr\'{e}d\'{e}ric and Kodate, Takako and P\'{e}rennes, St\'{e}phane},
  title =	{{Directed Grabbing Games or How to Politely Grab the Maximum Number of Olives in a Reception}},
  booktitle =	{13th International Conference on Fun with Algorithms (FUN 2026)},
  pages =	{8:1--8:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-417-8},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{366},
  editor =	{Iacono, John},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FUN.2026.8},
  URN =		{urn:nbn:de:0030-drops-257273},
  doi =		{10.4230/LIPIcs.FUN.2026.8},
  annote =	{Keywords: grabbing games, paths, directed grids}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2025.23
Parallel Complexity of Depth-First-Search and Maximal Path in Restricted Graph Classes

Authors: Archit Chauhan, Samir Datta, and M. Praveen

Published in: LIPIcs, Volume 360, 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)

Abstract
Constructing a Depth First Search (DFS) tree is a fundamental graph problem, whose parallel complexity is still not settled. Reif showed parallel intractability of lex-first DFS. In contrast, randomized parallel algorithms (and more recently, deterministic quasipolynomial parallel algorithms) are known for constructing a DFS tree in general (di)graphs. However a deterministic parallel algorithm for DFS in general graphs remains an elusive goal. Working towards this, a series of works gave deterministic NC algorithms for DFS in planar graphs and digraphs. We further extend these results to more general graph classes, by providing NC algorithms for (di)graphs of bounded genus, and for undirected H-minor-free graphs where H is a fixed graph with at most one crossing. For the case of (di)graphs of bounded treewidth, we further improve the complexity to a Logspace bound. Constructing a maximal path is a simpler problem (that reduces to DFS) for which no deterministic parallel bounds are known for general graphs. For planar graphs a bound of O(log n) parallel time on a CRCW PRAM (thus in NC²) is known. We improve this bound to Logspace.
Cite as

Archit Chauhan, Samir Datta, and M. Praveen. Parallel Complexity of Depth-First-Search and Maximal Path in Restricted Graph Classes. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 23:1-23:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chauhan_et_al:LIPIcs.FSTTCS.2025.23,
  author =	{Chauhan, Archit and Datta, Samir and Praveen, M.},
  title =	{{Parallel Complexity of Depth-First-Search and Maximal Path in Restricted Graph Classes}},
  booktitle =	{45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025)},
  pages =	{23:1--23:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-406-2},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{360},
  editor =	{Aiswarya, C. and Mehta, Ruta and Roy, Subhajit},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2025.23},
  URN =		{urn:nbn:de:0030-drops-251041},
  doi =		{10.4230/LIPIcs.FSTTCS.2025.23},
  annote =	{Keywords: Parallel Complexity, Graph Algorithms, Depth First Search, Maximal Path, Planar Graphs, Minor-Free, Treewidth, Logspace}
}
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
Poster Abstract
DOI: 10.4230/LIPIcs.GD.2025.47
Reconfigurations of Plane Caterpillars and Paths (Poster Abstract)

Authors: Todor Antić, Guillermo Gamboa Quintero, and Jelena Glišić

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

Abstract
Let S be a point set in the plane, and let 𝒫(S) and 𝒞(S) be the sets of all plane spanning paths and caterpillars on S. We study reconfiguration operations on 𝒫(S) and 𝒞(S). In particular, we prove that all of the commonly studied reconfigurations on plane spanning trees still yield connected reconfiguration graphs for caterpillars when S is in convex position. If S is in general position, we show that the rotation, compatible flip and flip graphs of 𝒞(S) are connected while the slide graph is sometimes disconnected, but always has a component of size 1/4(3ⁿ-1). We then study sizes of connected components in reconfiguration graphs of plane spanning paths. In this direction, we show that no component of size at most 7 can exist in the flip graph on 𝒫(S).
Cite as

Todor Antić, Guillermo Gamboa Quintero, and Jelena Glišić. Reconfigurations of Plane Caterpillars and Paths (Poster Abstract). In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 47:1-47:5, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{antic_et_al:LIPIcs.GD.2025.47,
  author =	{Anti\'{c}, Todor and Gamboa Quintero, Guillermo and Gli\v{s}i\'{c}, Jelena},
  title =	{{Reconfigurations of Plane Caterpillars and Paths}},
  booktitle =	{33rd International Symposium on Graph Drawing and Network Visualization (GD 2025)},
  pages =	{47:1--47:5},
  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.2025.47},
  URN =		{urn:nbn:de:0030-drops-250337},
  doi =		{10.4230/LIPIcs.GD.2025.47},
  annote =	{Keywords: reconfiguration graph, caterpillar, path, geometric graph}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.15
A Polynomial-Time Approximation Algorithm for Complete Interval Minors

Authors: Romain Bourneuf, Julien Cocquet, Chaoliang Tang, and Stéphan Thomassé

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
As shown by Robertson and Seymour, deciding whether the complete graph K_t is a minor of an input graph G is a fixed parameter tractable problem when parameterized by t. From the approximation viewpoint, a substantial gap remains: there is no PTAS for finding the largest complete minor unless P = NP, whereas the best known result is a polytime O(√ n)-approximation algorithm by Alon, Lingas and Wahlén. We investigate the complexity of finding K_t as interval minor in ordered graphs (i.e. graphs with a linear order on the vertices, in which intervals are contracted to form minors). Our main result is a polytime f(t)-approximation algorithm, where f is triply exponential in t but independent of n. The algorithm is based on delayed decompositions and shows that ordered graphs without a K_t interval minor can be constructed via a bounded number of three operations: closure under substitutions, edge union, and concatenation of a stable set. As a byproduct, graphs avoiding K_t as an interval minor have bounded chromatic number.
Cite as

Romain Bourneuf, Julien Cocquet, Chaoliang Tang, and Stéphan Thomassé. A Polynomial-Time Approximation Algorithm for Complete Interval Minors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bourneuf_et_al:LIPIcs.APPROX/RANDOM.2025.15,
  author =	{Bourneuf, Romain and Cocquet, Julien and Tang, Chaoliang and Thomass\'{e}, St\'{e}phan},
  title =	{{A Polynomial-Time Approximation Algorithm for Complete Interval Minors}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{15:1--15:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.15},
  URN =		{urn:nbn:de:0030-drops-243814},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.15},
  annote =	{Keywords: Approximation algorithm, Ordered graphs, Interval minors, Delayed decompositions}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.47
Shelling and Sinking Graphs on the Sphere

Authors: Jeff Erickson and Christian Howard

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

Abstract
We describe a promising approach to efficiently morph spherical graphs, extending earlier approaches of Awartani and Henderson [Trans. AMS 1987] and Kobourov and Landis [JGAA 2006]. Specifically, we describe two methods to morph shortest-path triangulations of the sphere by moving their vertices along longitudes into the southern hemisphere; we call a triangulation sinkable if such a morph exists. Our first method generalizes a longitudinal shelling construction of Awartani and Henderson; a triangulation is sinkable if a specific orientation of its dual graph is acyclic. We describe a simple polynomial-time algorithm to find a longitudinally shellable rotation of a given spherical triangulation, if one exists; we also construct a spherical triangulation that has no longitudinally shellable rotation. Our second method is based on a linear-programming characterization of sinkability. By identifying its optimal basis, we show that this linear program can be solved in O(n^{ω/2}) time, where ω is the matrix-multiplication exponent, assuming the underlying linear system is non-singular. Finally, we pose several conjectures and describe experimental results that support them.
Cite as

Jeff Erickson and Christian Howard. Shelling and Sinking Graphs on the Sphere. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 47:1-47:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{erickson_et_al:LIPIcs.SoCG.2025.47,
  author =	{Erickson, Jeff and Howard, Christian},
  title =	{{Shelling and Sinking Graphs on the Sphere}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{47:1--47:18},
  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.2025.47},
  URN =		{urn:nbn:de:0030-drops-231996},
  doi =		{10.4230/LIPIcs.SoCG.2025.47},
  annote =	{Keywords: morphing, planar graphs, spherical graph drawing, longitudinal shelling}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2025.75
Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers

Authors: Elizaveta Streltsova and Uli Wagner

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

Abstract
A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V. Moreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level). To prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope.
Cite as

Elizaveta Streltsova and Uli Wagner. Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 75:1-75:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{streltsova_et_al:LIPIcs.SoCG.2025.75,
  author =	{Streltsova, Elizaveta and Wagner, Uli},
  title =	{{Levels in Arrangements: Linear Relations, the g-Matrix, and Applications to Crossing Numbers}},
  booktitle =	{41st International Symposium on Computational Geometry (SoCG 2025)},
  pages =	{75:1--75:18},
  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.2025.75},
  URN =		{urn:nbn:de:0030-drops-232276},
  doi =		{10.4230/LIPIcs.SoCG.2025.75},
  annote =	{Keywords: Levels in arrangements, k-sets, k-facets, convex polytopes, f-vector, h-vector, g-vector, Dehn-Sommerville relations, Radon partitions, Gale duality, g-matrix}
}
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.2020.62
Barycentric Cuts Through a Convex Body

Authors: Zuzana Patáková, Martin Tancer, and Uli Wagner

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)

Abstract
Let K be a convex body in ℝⁿ (i.e., a compact convex set with nonempty interior). Given a point p in the interior of K, a hyperplane h passing through p is called barycentric if p is the barycenter of K ∩ h. In 1961, Grünbaum raised the question whether, for every K, there exists an interior point p through which there are at least n+1 distinct barycentric hyperplanes. Two years later, this was seemingly resolved affirmatively by showing that this is the case if p=p₀ is the point of maximal depth in K. However, while working on a related question, we noticed that one of the auxiliary claims in the proof is incorrect. Here, we provide a counterexample; this re-opens Grünbaum’s question. It follows from known results that for n ≥ 2, there are always at least three distinct barycentric cuts through the point p₀ ∈ K of maximal depth. Using tools related to Morse theory we are able to improve this bound: four distinct barycentric cuts through p₀ are guaranteed if n ≥ 3.
Cite as

Zuzana Patáková, Martin Tancer, and Uli Wagner. Barycentric Cuts Through a Convex Body. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{patakova_et_al:LIPIcs.SoCG.2020.62,
  author =	{Pat\'{a}kov\'{a}, Zuzana and Tancer, Martin and Wagner, Uli},
  title =	{{Barycentric Cuts Through a Convex Body}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{62:1--62:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.62},
  URN =		{urn:nbn:de:0030-drops-122201},
  doi =		{10.4230/LIPIcs.SoCG.2020.62},
  annote =	{Keywords: convex body, barycenter, Tukey depth, smooth manifold, critical points}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2019.39
Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices

Authors: Radoslav Fulek and Jan Kynčl

Published in: LIPIcs, Volume 129, 35th International Symposium on Computational Geometry (SoCG 2019)

Abstract
The genus g(G) of a graph G is the minimum g such that G has an embedding on the orientable surface M_g of genus g. A drawing of a graph on a surface is independently even if every pair of nonadjacent edges in the drawing crosses an even number of times. The Z_2-genus of a graph G, denoted by g_0(G), is the minimum g such that G has an independently even drawing on M_g. By a result of Battle, Harary, Kodama and Youngs from 1962, the graph genus is additive over 2-connected blocks. In 2013, Schaefer and Štefankovič proved that the Z_2-genus of a graph is additive over 2-connected blocks as well, and asked whether this result can be extended to so-called 2-amalgamations, as an analogue of results by Decker, Glover, Huneke, and Stahl for the genus. We give the following partial answer. If G=G_1 cup G_2, G_1 and G_2 intersect in two vertices u and v, and G-u-v has k connected components (among which we count the edge uv if present), then |g_0(G)-(g_0(G_1)+g_0(G_2))|<=k+1. For complete bipartite graphs K_{m,n}, with n >= m >= 3, we prove that g_0(K_{m,n})/g(K_{m,n})=1-O(1/n). Similar results are proved also for the Euler Z_2-genus. We express the Z_2-genus of a graph using the minimum rank of partial symmetric matrices over Z_2; a problem that might be of independent interest.
Cite as

Radoslav Fulek and Jan Kynčl. Z_2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 39:1-39:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{fulek_et_al:LIPIcs.SoCG.2019.39,
  author =	{Fulek, Radoslav and Kyn\v{c}l, Jan},
  title =	{{Z\underline2-Genus of Graphs and Minimum Rank of Partial Symmetric Matrices}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{39:1--39:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-104-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{129},
  editor =	{Barequet, Gill and Wang, Yusu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2019.39},
  URN =		{urn:nbn:de:0030-drops-104439},
  doi =		{10.4230/LIPIcs.SoCG.2019.39},
  annote =	{Keywords: graph genus, minimum rank of a partial matrix, Hanani-Tutte theorem, graph amalgamation}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2018.39
Hanani-Tutte for Approximating Maps of Graphs

Authors: Radoslav Fulek and Jan Kyncl

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)

Abstract
We resolve in the affirmative conjectures of A. Skopenkov and Repovs (1998), and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the setting of approximating maps of graphs on 2-dimensional surfaces by embeddings. Our proof of this result is constructive and almost immediately implies an efficient algorithm for testing whether a given piecewise linear map of a graph in a surface is approximable by an embedding. More precisely, an instance of this problem consists of (i) a graph G whose vertices are partitioned into clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a region R of a 2-dimensional compact surface M given as the union of a set of pairwise disjoint discs corresponding to the clusters and a set of pairwise disjoint "pipes" corresponding to the bundles, connecting certain pairs of these discs. We are to decide whether G can be embedded inside M so that the vertices in every cluster are drawn in the corresponding disc, the edges in every bundle pass only through its corresponding pipe, and every edge crosses the boundary of each disc at most once.
Cite as

Radoslav Fulek and Jan Kyncl. Hanani-Tutte for Approximating Maps of Graphs. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fulek_et_al:LIPIcs.SoCG.2018.39,
  author =	{Fulek, Radoslav and Kyncl, Jan},
  title =	{{Hanani-Tutte for Approximating Maps of Graphs}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{39:1--39:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.39},
  URN =		{urn:nbn:de:0030-drops-87527},
  doi =		{10.4230/LIPIcs.SoCG.2018.39},
  annote =	{Keywords: Hanani-Tutte theorem, graph embedding, map approximation, weak embedding, clustered planarity}
}
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