On grids in topological graphs
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Fox_On grids.pdf
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256.13 KB
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Author(s)
Ackerman, Eyal
Fox, Jacob
Pach, János
Suk, Andrew
Date Issued
February 2014
Journal
Computational Geometry
Publisher
Elsevier
Citation
Ackerman, Eyal, Jacob Fox, János Pach, and Andrew Suk. “On Grids in Topological Graphs.” Computational Geometry 47, no. 7 (August 2014): 710–723.
Version
Author's final manuscript
Abstract
A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straight-line segment, then G is called a geometric graph. 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 every n-vertex topological graph with no k-grid has O[subscript k](n) edges. We conjecture that the number of edges of every n-vertex topological graph with no k-grid such that all of its 2k edges have distinct endpoints is O[subscript k(n). This conjecture is shown to be true apart from an iterated logarithmic factor ⁎. A k-grid is natural if its edges have distinct endpoints, and the arcs representing each of its edge subsets are pairwise disjoint. We also conjecture that every n-vertex geometric graph with no natural k-grid has edges, but we can establish only an O[subscript k](nlog[superscript 2] n) upper bound. We verify the above conjectures in several special cases.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-NonCommercial-NoDerivs License
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.comgeo.2014.02.003
