Quasi-optimal range searching in spaces of finite VC-dimension
Discrete & Computational Geometry - Selected papers from the fourth ACM symposium on computational geometry, Univ. of Illinois, Urbana-Champaign, June 6 8, 1988
Approximation schemes for covering and packing problems in image processing and VLSI
Journal of the ACM (JACM)
Empty triangles in drawings of the complete graph
Discrete Mathematics
Label placement by maximum independent set in rectangles
WADS '97 Selected papers presented at the international workshop on Algorithms and data structure
Journal of Combinatorial Theory Series A
On empty convex polygons in a planar point set
Journal of Combinatorial Theory Series A
Empty Convex Hexagons in Planar Point Sets
Discrete & Computational Geometry
Disjoint edges in topological graphs
IJCCGGT'03 Proceedings of the 2003 Indonesia-Japan joint conference on Combinatorial Geometry and Graph Theory
Disjoint edges in complete topological graphs
Proceedings of the twenty-eighth annual symposium on Computational geometry
Density theorems for intersection graphs of t-monotone curves
GD'12 Proceedings of the 20th international conference on Graph Drawing
| Hi-index | 0.00 |
A simple topological graph is a graph drawn in the plane so that its edges are represented by continuous arcs with the property that any two of them meet at most once. We present a novel tool for finding crossing free subgraphs in simple topological graphs. Using this tool, we solve the following two problems. Let G be a complete simple topological graph on n vertices. The three edges induced by any triplet of vertices in G form a simple closed curve. If this curve contains no vertex in its interior (exterior), then we say that the triplet forms an empty triangle. In 1998, Harborth proved that G has at least 2 empty triangles, and he conjectured that the number of empty triangles is at least 2n/3. We settle Harborth's conjecture in the affirmative. We also present a new proof of a result by Suk stating that every complete simple topological graph on n vertices contains Ω(n1/3) pairwise disjoint edges.