Halving Lines and Their Underlying Graphs

  title={Halving Lines and Their Underlying Graphs},
  author={Tanya Khovanova and Dai Yang},
In this paper we study underlying graphs corresponding to a set of halving lines. We establish many properties of such graphs. In addition, we tighten the upper bound for the number of halving lines. 
Connected Components of Underlying Graphs of Halving Lines
The connected components of underlying graphs of halving lines' configurations are discussed and it is proved that every connected component of the underlying graph is itself an underlying graph.
Fission of Halving Edges Graphs
In this paper we discuss an operation on halving edges graph that we call fission. Fission replaces each point in a given configuration with a small cluster of k points. The operation interacts
An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges
The lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2 is improved.


Graphs drawn with few crossings per edge
It is shown that if a graph ofv vertices can be drawn in the plane so that every edge crosses at mostk>0 others, then its number of edges cannot exceed 4.108√kv, and a better bound is established, (k+3)(v−2), which is tight fork=1 and 2.
Point sets with many k-sets
  • G. Tóth
  • Mathematics, Computer Science
    SCG '00
  • 2000
This paper improves the bounds of Erdős, Lovász, et al. on the number of halving hyperplanes in higher dimensions by constructing a set of n points in the plane with ne-Omega k -sets.
Improved Bounds for Planar k -Sets and Related Problems
  • T. Dey
  • Mathematics
    Discret. Comput. Geom.
  • 1998
This is the first considerable improvement on this bound after its early solution approximately 27 years ago and applies to improve the current bounds on the combinatorial complexities of k -levels in the arrangement of line segments, convex polygons in the union of n lines, parametric minimum spanning trees, and parametric matroids in general.
Wikipedia article on Pseudoforests available at
  • Wikipedia article on Pseudoforests available at
  • 2012
Gráfok elő́ırt fokszámú pontokkal (in Hungarian)
  • Matematikai Lapok
  • 1960
Online Encyclopedia of Integer Sequences (OEIS)
  • Online Encyclopedia of Integer Sequences (OEIS)
  • 2012
Gráfok el˝ oírt fokszámú pontokkal
  • Matematikai Lapok
  • 1960