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DOI: 10.4230/LIPIcs.SoCG.2019.14

Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12

Authors: Drago Bokal, Zdeněk Dvořák, Petr Hliněný, Jesús Leaños, Bojan Mohar, and Tilo Wiedera

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

Abstract

We study c-crossing-critical graphs, which are the minimal graphs that require at least c edge-crossings when drawn in the plane. For every fixed pair of integers with c >= 13 and d >= 1, we give first explicit constructions of c-crossing-critical graphs containing a vertex of degree greater than d. We also show that such unbounded degree constructions do not exist for c <=12, precisely, that there exists a constant D such that every c-crossing-critical graph with c <=12 has maximum degree at most D. Hence, the bounded maximum degree conjecture of c-crossing-critical graphs, which was generally disproved in 2010 by Dvořák and Mohar (without an explicit construction), holds true, surprisingly, exactly for the values c <=12.

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Drago Bokal, Zdeněk Dvořák, Petr Hliněný, Jesús Leaños, Bojan Mohar, and Tilo Wiedera. Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 14:1-14:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bokal_et_al:LIPIcs.SoCG.2019.14,
  author =	{Bokal, Drago and Dvo\v{r}\'{a}k, Zden\v{e}k and Hlin\v{e}n\'{y}, Petr and Lea\~{n}os, Jes\'{u}s and Mohar, Bojan and Wiedera, Tilo},
  title =	{{Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c \langle= 12}},
  booktitle =	{35th International Symposium on Computational Geometry (SoCG 2019)},
  pages =	{14:1--14:15},
  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.2019.14},
  URN =		{urn:nbn:de:0030-drops-104183},
  doi =		{10.4230/LIPIcs.SoCG.2019.14},
  annote =	{Keywords: graph drawing, crossing number, crossing-critical, zip product}
}

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DOI: 10.4230/LIPIcs.SOCG.2015.240

Finding All Maximal Subsequences with Hereditary Properties

Authors: Drago Bokal, Sergio Cabello, and David Eppstein

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract

Consider a sequence s_1,...,s_n of points in the plane. We want to find all maximal subsequences with a given hereditary property P: find for all indices i the largest index j^*(i) such that s_i,...,s_{j^*(i)} has property P. We provide a general methodology that leads to the following specific results: - In O(n log^2 n) time we can find all maximal subsequences with diameter at most 1. - In O(n log n loglog n) time we can find all maximal subsequences whose convex hull has area at most 1. - In O(n) time we can find all maximal subsequences that define monotone paths in some (subpath-dependent) direction. The same methodology works for graph planarity, as follows. Consider a sequence of edges e_1,...,e_n over a vertex set V. In O(n log n) time we can find, for all indices i, the largest index j^*(i) such that (V,{e_i,..., e_{j^*(i)}}) is planar.

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Drago Bokal, Sergio Cabello, and David Eppstein. Finding All Maximal Subsequences with Hereditary Properties. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 240-254, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{bokal_et_al:LIPIcs.SOCG.2015.240,
  author =	{Bokal, Drago and Cabello, Sergio and Eppstein, David},
  title =	{{Finding All Maximal Subsequences with Hereditary Properties}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{240--254},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.240},
  URN =		{urn:nbn:de:0030-drops-51132},
  doi =		{10.4230/LIPIcs.SOCG.2015.240},
  annote =	{Keywords: convex hull, diameter, monotone path, sequence of points, trajectory}
}

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Bounded Degree Conjecture Holds Precisely for c-Crossing-Critical Graphs with c <= 12

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Finding All Maximal Subsequences with Hereditary Properties

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