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On grids in topological graphs

Authors:

Andrew SukAuthors Info & Claims

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

Pages 403 - 412

https://doi.org/10.1145/1542362.1542430

Published: 08 June 2009 Publication History

Abstract

A topological graph is a graph drawn in the plane with vertices represented by points and edges as arcs connecting its vertices. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges.

We conjecture that this statement remains true (1) for topological graphs in which only k-grids consisting of 2k vertex-disjoint edges are forbidden, and (2) for graphs drawn by straight-line edges, with no k-element sets of edges such that every edge in the first set crosses every edge in the other set and each pair of edges within the same set is disjoint.

These conjectures are shown to be true apart from log* n and log² n factors, respectively. We also settle the conjectures for some special cases.

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Cited By

Fox JPach JDey TWhitesides S(2012)String graphs and incomparability graphsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261311(405-414)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261311
Fox JPach J(2012)String graphs and incomparability graphsAdvances in Mathematics10.1016/j.aim.2012.03.011230:3(1381-1401)Online publication date: Jul-2012
https://doi.org/10.1016/j.aim.2012.03.011
Pach JRadoičić RTóth G(2012)Tangled ThracklesComputational Geometry10.1007/978-3-642-34191-5_4(45-53)Online publication date: 2012
https://doi.org/10.1007/978-3-642-34191-5_4
Show More Cited By

Index Terms

On grids in topological graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Enumeration

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cover image ACM Conferences

SCG '09: Proceedings of the twenty-fifth annual symposium on Computational geometry

June 2009

426 pages

ISBN:9781605585017

DOI:10.1145/1542362

Program Chairs:
John Hershberger
Mentor Graphics Corp.
,
Efi Fogel
Tel Aviv University

Copyright © 2009 ACM.

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Published: 08 June 2009

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SoCG '09: 25th Annual Symposium on Computational Geometry

June 8 - 10, 2009

Aarhus, Denmark

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Cited By

Fox JPach JDey TWhitesides S(2012)String graphs and incomparability graphsProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261311(405-414)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261311
Fox JPach J(2012)String graphs and incomparability graphsAdvances in Mathematics10.1016/j.aim.2012.03.011230:3(1381-1401)Online publication date: Jul-2012
https://doi.org/10.1016/j.aim.2012.03.011
Pach JRadoičić RTóth G(2012)Tangled ThracklesComputational Geometry10.1007/978-3-642-34191-5_4(45-53)Online publication date: 2012
https://doi.org/10.1007/978-3-642-34191-5_4
Suk A(2011)k-Quasi-planar graphsProceedings of the 19th international conference on Graph Drawing10.1007/978-3-642-25878-7_26(266-277)Online publication date: 21-Sep-2011
https://dl.acm.org/doi/10.1007/978-3-642-25878-7_26

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