# Edge complexity of geometric graphs on convex independent point sets

@article{Khopkar2016EdgeCO, title={Edge complexity of geometric graphs on convex independent point sets}, author={A. Khopkar}, journal={arXiv: Discrete Mathematics}, year={2016} }

In this paper, we focus on a generalised version of Gabriel graphs known as Locally Gabriel graphs ($LGGs$) and Unit distance graphs ($UDGs$) on convexly independent point sets. $UDGs$ are sub graphs of $LGGs$. We give a simpler proof for the claim that $LGGs$ on convex independent point sets have $2n \log n + O(n)$ edges. Then we prove that unit distance graphs on convex independent point sets have $O(n)$ edges improving the previous known bound of $O(n \log n)$.

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SHOWING 1-10 OF 37 REFERENCES

Turán-type results for partial orders and intersection graphs of convex sets

- Mathematics
- 2010

We prove Ramsey-type results for intersection graphs of geometric objects. In particular, we prove the following bounds, all of which are tight apart from the constant c. There is a constant c > 0… Expand

Graph Drawings with no k Pairwise Crossing Edges

- Mathematics, Computer Science
- Graph Drawing
- 1997

A new, simpler proof of the bound that, for any fixed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges is given. Expand

A Turán-type Extremal Theory of Convex Geometric Graphs

- Mathematics
- 2003

We study Turan-type extremal questions for graphs with an additional cyclic ordering of the vertices, i.e. for convex geometric graphs. If a suitably defined chromatic number of the excluded subgraph… Expand

On Locally Gabriel Geometric Graphs

- Mathematics, Computer Science
- Graphs Comb.
- 2015

Let $$P$$P be a set of $$n$$n points in the plane. A geometric graph $$G$$G on $$P$$P is said to be locally Gabriel if for every edge $$(u,v)$$(u,v) in $$G$$G, the Euclidean disk with the segment… Expand

Ramsey-Type Results for Geometric Graphs II

- Computer Science
- Symposium on Computational Geometry
- 1997

It is shown that for any 2{coloring of the n2 segments determined by n points in the plane, one of the color classes contains non-crossing cycles of lengths 3; 4; : : : ; bqn=2c, and it is proved that there is a non-Crossing path of length (n2=3), all of whose edges are of the same color. Expand

The Unit Distance Problem for Centrally Symmetric Convex Polygons

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 2002

It is proved that fsym(n) is the restriction of f( n) to centrally symmetric convex n -gons and that fK(n), the maximum number of unit segments spanned by n points in the boundary of K, Whenever K is centrally asymmetric or has width >1, is given. Expand

Proximity Structures for Geometric Graphs

- Mathematics, Computer Science
- Int. J. Comput. Geom. Appl.
- 2010

This paper defines Voronoi diagrams, Delaunay triangulations, relative neighborhood graphs, Gabriel graphs which are related to the graph structure and study their complexities when G is a general geometric graph or G is some special graph derived from the application area of wireless networks. Expand

Crossing Numbers and Hard Erdős Problems in Discrete Geometry

- 1997

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… Expand

Extremal theory for convex matchings in convex geometric graphs

- Mathematics, Computer Science
- Discret. Comput. Geom.
- 1996

A convex geometric graphG of ordern consists of the set of vertices of a plane convexn-gonP together with some edges, and/or diagonals ofP as edges and the definition of “l-free” is removed. Expand

Crossing Numbers and Hard Erd} os Problems in Discrete Geometry

- Mathematics
- 1997

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… Expand