dc.contributor.author | Ackerman, Eyal | |
dc.contributor.author | Fox, Jacob | |
dc.contributor.author | Pach, János | |
dc.contributor.author | Suk, Andrew | |
dc.date.accessioned | 2017-07-11T17:22:57Z | |
dc.date.available | 2017-07-11T17:22:57Z | |
dc.date.issued | 2014-02 | |
dc.date.submitted | 2009-12 | |
dc.identifier.issn | 09257721 | |
dc.identifier.uri | http://hdl.handle.net/1721.1/110635 | |
dc.description.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. | en_US |
dc.language.iso | en_US | |
dc.publisher | Elsevier | en_US |
dc.relation.isversionof | http://dx.doi.org/10.1016/j.comgeo.2014.02.003 | en_US |
dc.rights | Creative Commons Attribution-NonCommercial-NoDerivs License | en_US |
dc.rights.uri | http://creativecommons.org/licenses/by-nc-nd/4.0/ | en_US |
dc.source | Other univ. web domain | en_US |
dc.title | On grids in topological graphs | en_US |
dc.type | Article | en_US |
dc.identifier.citation | Ackerman, Eyal, Jacob Fox, János Pach, and Andrew Suk. “On Grids in Topological Graphs.” Computational Geometry 47, no. 7 (August 2014): 710–723. | en_US |
dc.contributor.department | Massachusetts Institute of Technology. Department of Mathematics | en_US |
dc.contributor.mitauthor | Fox, Jacob | |
dc.contributor.mitauthor | Suk, Andrew | |
dc.relation.journal | Computational Geometry | en_US |
dc.eprint.version | Author's final manuscript | en_US |
dc.type.uri | http://purl.org/eprint/type/JournalArticle | en_US |
eprint.status | http://purl.org/eprint/status/PeerReviewed | en_US |
dspace.orderedauthors | Ackerman, Eyal; Fox, Jacob; Pach, János; Suk, Andrew | en_US |
dspace.embargo.terms | N | en_US |
mit.license | PUBLISHER_CC | en_US |
mit.metadata.status | Complete | |