On grids in topological graphs

Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 log 2 n factors, respectively. We also settle the conjectures for some special cases.

Original languageEnglish
Title of host publicationProceedings of the 25th Annual Symposium on Computational Geometry, SCG'09
Number of pages10
StatePublished - 2009
Externally publishedYes
Event25th Annual Symposium on Computational Geometry, SCG'09 - Aarhus, Denmark
Duration: 8 Jun 200910 Jun 2009

Publication series

NameProceedings of the Annual Symposium on Computational Geometry


Conference25th Annual Symposium on Computational Geometry, SCG'09


  • Geometric graphs
  • Topological graphs
  • Turán-type problems

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Computational Mathematics


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