# Note on Geometric Graphs

@article{Tth2000NoteOG, title={Note on Geometric Graphs}, author={G{\'e}za T{\'o}th}, journal={J. Comb. Theory, Ser. A}, year={2000}, volume={89}, pages={126-132} }

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position and the edges are represented by straight line segments connecting the corresponding points. We show that a geometric graph of n vertices with no k+1 pairwise disjoint edges has at most 29k2n edges.

## 39 Citations

### Geometric Graphs with No Three Disjoint Edges

- MathematicsDiscret. Comput. Geom.
- 2005

It is shown that a geometric graph on n vertices with no three pairwise disjoint edges has at most 2.5n edges, which is tight up to an additive constant.

### Disjoint Edges in Topological Graphs

- MathematicsIJCCGGT
- 2003

It is shown that the maximum number of edges of a simple topological graph with n vertices and no k pairwise disjoint edges is O(nlog4k−8n) edges.

### Edge-Removal and Non-Crossing Configurations in Geometric Graphs

- MathematicsDiscret. Math. Theor. Comput. Sci.
- 2010

This work studies the following extremal problem for geometric graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the remaining graph still contains a certain non-crossing subgraph.

### The Beginnings of Geometric Graph Theory

- Mathematics
- 2013

Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straight-line edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz…

### Unavoidable patterns in complete simple topological graphs

- Mathematics
- 2022

In this paper, we show that every complete n -vertex simple topological graph contains a topological subgraph on at least (log n ) 1 / 4 − o (1) vertices that is weakly isomorphic to the complete…

### On grids in topological graphs

- MathematicsSCG '09
- 2009

This work conjectures that for a fixed constant k, every n-vertex topological graph with no k-grid has O(n) edges and shows these conjectures to be true apart from log* n and log2 n factors.

### Matchings in Geometric Graphs

- Mathematics, Computer ScienceArXiv
- 2016

This thesis studies matching problems in various geometric graphs and establishes lower and upper bounds on the size of different kinds of matchings in various geometry graphs, and presents algorithms for computing such matchings.

### Geometric Intersection Patterns and the Theory of Topological Graphs 3 Using similar ideas

- Mathematics
- 2014

The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown…

## References

SHOWING 1-10 OF 11 REFERENCES

### Geometric Graphs with Few Disjoint Edges

- MathematicsSCG '98
- 1998

Abstract. A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the…

### Disjoint edges in geometric graphs

- MathematicsDiscret. Comput. Geom.
- 1989

It is shown that any configuration consisting of a setV ofn points in general position in the plane and a set of 6n − 5 closed straight line segments whose endpoints lie inV, contains three pairwise disjoint line segments.

### Some geometric applications of Dilworth’s theorem

- MathematicsDiscret. Comput. Geom.
- 1994

It is shown that every geometric graph withn vertices andm>k4n edges containsk+1 pairwise disjoint edges, and it is proved that, given a set of pointsV and aSet of axis-parallel rectangles in the plane, then either there arek-1 rectangles such that no point ofV belongs to more than one of them, or the authors can find an at most 2·105k8 element subset ofV meeting all rectangles.

### Some geometric applications of Dilworth's theorem

- MathematicsSCG '93
- 1993

Bothproofs are based on Dilworth's theorem on partially ordered sets and improves a result of Ding, Seymour and Winkler.

### Combinatorial geometry

- MathematicsWiley-Interscience series in discrete mathematics and optimization
- 1995

### A DECOMPOSITION THEOREM FOR PARTIALLY ORDERED SETS

- Mathematics
- 1950

Otherwise a and b are non-comparable. A subset S of P is independent if every two distinct elements of S are non-comparable. S is dependent if it contains two distinct elements which are comparable.…

### A Ramsey-type result for convex sets

- Mathematics, Economics
- 1994

Keywords: convex compact sets Note: Professor Pach's number: [096] Reference DCG-ARTICLE-1994-004doi:10.1112/blms/26.2.132 Record created on 2008-11-14, modified on 2017-05-12

### Jahresbericht der deutschen Mathematiker-Vereinigung

- 1902

Vorwort zu Heft 1-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Vorwort zu Heft 2-10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .…

### Törőcsik, A Ramsey-type result for planar convex sets, The Bulletin of the London

- Mathematical Society
- 1994