Discrete and Computational Geometry

  title={Discrete and Computational Geometry},
  author={Jan van Leeuwen and Jin Akiyama and Mikio Kano and Masatsugu Urabe and Jan van Leeuwen},
  booktitle={Lecture Notes in Computer Science},
This talk surveys how geometric information can be effectively used for efficient algorithms with focus on clustering problems. Given a complete weighted graph G of n vertices, is there a partition of the vertex set into k disjoint subsets so that the maximum weight of an innercluster edge (whose two endpoints both belong to the same subset) is minimized. This problem is known to be NP-complete even for k = 3. The case of k = 2, that is, bipartition problem is solvable in polynomial time. On… 
1 Citations
Equal Area Polygons in Convex Bodies
It is shown that for a convex quadrilateral K of area 1, there exist n internally disjoint triangles of equal area such that the sum of their areas is at least $\frac {4n}{4n+1}$.


Crossing Numbers and Hard Erdős Problems in Discrete Geometry
We show that an old but not well-known lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the
On Geometric Graphs with No k Pairwise Parallel Edges
  • P. Valtr
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1998
It is shown that, for any fixed k ≥ 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graphs on n n verticeswith no k -1 pairwise crossing edges containing at mostO(n log n) edges.
New Bounds on Crossing Numbers
The crossing number, cr(G), of a graph G is the least number of crossing points in any drawing of G in the plane. Denote by κ (n, e) the minimum of cr(G) taken over all graphs with n vertices and at
A Separator Theorem for Planar Graphs
Let G be any n-vertex planar graph. We prove that the vertices of G can be partitioned into three sets A, B, C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more
Which Crossing Number Is It Anyway?
A drawing of a graph G is a mapping which assigns to each vertex a point of the plane and to each edge a simple continuous arc connecting the corresponding two points. The crossing number of G is the
Extremal problems in discrete geometry
Several theorems involving configurations of points and lines in the Euclidean plane are established, including one that shows that there is an absolute constantc3 so that whenevern points are placed in the plane not all on the same line, then there is one point on more thanc3n of the lines determined by then points.
Graphs Drawn with Few Crossings Per Edge
It is shown that if a graph of v vertices can be drawn in the plane so that every edge crosses at most k> 0 others, then its number of edges cannot exceed 4, and a better bound is established, (k + 3)(u− 2), which is tight for k=1 and 2.
Improved Bounds for Planar k -Sets and Related Problems
  • T. Dey
  • Mathematics, Computer Science
    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.
Combinatorial complexity bounds for arrangements of curves and surfaces
  • K. Clarkson, M. Sharir
  • Mathematics, Computer Science
    [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
  • 1988
The authors study both the incidence counting and the many-faces problem for various kinds of curves, including lines, pseudolines, unit circles, general circles, and pseudocircles. They also extend
Crossing Number is NP-Complete
In this paper we consider a problem related to questions of optimal circuit layout: Given a graph or network, how can we embed it in a planar surface so as to minimize the number of edge-crossings?