# Small sets supporting fary embeddings of planar graphs

@inproceedings{Fraysseix1988SmallSS, title={Small sets supporting fary embeddings of planar graphs}, author={Hubert de Fraysseix and J{\'a}nos Pach and Richard Pollack}, booktitle={STOC '88}, year={1988} }

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph with <italic>n</italic> vertices has a Fáry embedding (i.e., straight-line embedding) on the 2<italic>n</italic> - 4 by <italic>n</italic> - 2 grid and provide an &Ogr;(<italic>n</italic>) space, &Ogr;(<italic>n</italic> log <italic>n</italic>) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to…

## 203 Citations

A lower bound on the size of universal sets for planar graphs

- Computer ScienceSIGA
- 1989

It is shown that any set said to be a n-universal set has size at least 1.098, for sufficiently large planar graphs.

Separators for sphere-packings and nearest neighbor graphs

- MathematicsJACM
- 1997

This result implies that every triangulated planar graph is isomorphic to the intersection graph of a disk-packing, which gives a new geometric proof of the planar separator theorem of Lipton and Tarjan, but also generalizes it to higher dimensions.

Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs

- MathematicsWADS
- 2003

The properties of Schnyder’s realizers and canonical ordering trees of plane graphs are studied and compact drawings of two styles for any plane graph G with n vertices are obtained.

Canonical Ordering Trees and Their Applications in Graph Drawing

- MathematicsDiscret. Comput. Geom.
- 2005

The properties of Schnyder’s realizers and canonical ordering trees of plane graphs are studied to obtain compact drawings of two styles for any plane graph G with n vertices and it is shown that G has a visibility representation with height at most 15n/16 ⌉.

Connectivity check in 3-connected planar graphs with obstacles

- Mathematics, Computer ScienceElectron. Notes Discret. Math.
- 2008

Minimum-Width Grid Drawings of Plane Graphs

- Computer Science, MathematicsGraph Drawing
- 1994

This paper shows that this bound is tight, by presenting a grid drawing algorithm that produces drawings of width [2(n 1)/3J] and the height of the produced drawings is bounded by 4L2( n 1/3J 1.

Convex Grid Drawings of Planar Graphs\\with Constant Edge-Vertex Resolution

- MathematicsArXiv
- 2022

We continue the study of the area requirement of convex straight-line grid drawings of 3-connected plane graphs, which has been intensively investigated in the last decades. Motivated by…

Nice drawings of graphs are computationally hard

- Computer Science, MathematicsInformatics and Psychology Workshop
- 1988

A formal approach to this problem and from an algorithmic point of view optimal embeddings or equivalently nice drawings of graphs are intractable, which means that one must pay for nice drawings with a high computational effort.

The Complexity of Drawing Graphs on Few Lines and Few Planes

- Mathematics, Computer ScienceGraph Drawing
- 2016

This work investigates the problem of drawing graphs in 2D and 3D such that their edges (or only their vertices) can be covered by few lines or planes, and shows lower and upper bounds for the numbers of lines and planes needed for covering drawings of graphs in certain graph classes.

Embeddings of Polytopes and Polyhedral Complexes

- Mathematics
- 2012

Author(s): Wilson, Stedman | Advisor(s): Pak, Igor | Abstract: When does a topological polyhedral complex (embedded in Rd) admit a geometric realization (a rectilinear embedding in Rd)? What are the…

## References

SHOWING 1-4 OF 4 REFERENCES

Efficient Planarity Testing

- Computer ScienceJACM
- 1974

An efficient algorithm to determine whether an arbitrary graph G can be embedded in the plane is described, which used depth-first search and has time and space bounds.

Universality considerations in VLSI circuits

- Computer Science, MathematicsIEEE Transactions on Computers
- 1981

The problem of embedding the interconnection pattern of a circuit into a two-dimensional surface of minimal area is discussed and restricted classes of graphs have to be considered in order to achieve compact embeddings.

Slimming down search structures: A functional approach to algorithm design

- Computer Science26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
- 1985

New upper bounds on the complexity of several "rectangle" problems are established, including optimal algorithms for range counting and rectangle searching in two dimensions that involve linear space implementations of range trees and segment trees.