Intersection reverse sequences and geometric applications

Adam Marcus, Gábor Tardos

Research output: Contribution to journalArticle

33 Scopus citations

Abstract

Pinchasi and Radoičić [On the number of edges in geometric graphs with no self-intersecting cycle of length 4, in: J. Pach (Ed.), Towards a Theory of Geometric Graphs, Contemporary Mathematics, vol. 342, American Mathematical Society, Providence, RI, 2004] used the following observation to bound the number of edges of a topological graph without a self-crossing cycle of length 4: if we make a list of the neighbors for every vertex in such a graph and order these lists cyclically according to the order of the emanating edges, then the common elements in any two lists have reversed cyclic order. Building on their work we give an improved estimate on the size of the lists having this property. As a consequence we get that a topological graph on n vertices not containing a self-crossing C4 has O(n3/2 log n) edges. Our result also implies that n pseudo-circles in the plane can be cut into O(n3/2 log n) pseudo-segments, which in turn implies bounds on point-curve incidences and on the complexity of a level of an arrangement of curves.

Original languageEnglish (US)
Pages (from-to)675-691
Number of pages17
JournalJournal of Combinatorial Theory. Series A
Volume113
Issue number4
DOIs
StatePublished - May 1 2006

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Cyclic order
  • Extremal combinatorics
  • Pseudocircles
  • Topological graphs

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