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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).
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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.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}
}

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