# Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces

@article{Bokowski2018CombinatorialCQ, title={Combinatorial configurations, quasiline arrangements, and systems of curves on surfaces}, author={J{\"u}rgen Bokowski and Jurij Kovi{\vc} and Toma{\vz} Pisanski and Arjana Zitnik}, journal={Ars Math. Contemp.}, year={2018}, volume={14}, pages={97-116} }

It is well known that not every combinatorial configuration admits a geometric realization with points and lines. Moreover, some of them do not even admit realizations with pseudoline arrangements, i.e., they are not topological. In this paper we generalize the concept of topological configurations to a more general one (in a least possible way) such that every combinatorial configuration is realizable in this way. In particular, we generalize the notion of a pseudoline arrangement to the…

## 5 Citations

Arrangements of pseudocircles on surfaces

- MathematicsArXiv
- 2017

This paper shows that under an even more general definition of an arrangement, an arrangement of pseudocircles is embeddable into $\Sigma_g$ if and only if all of its subarrangements of size at most $4g+5$ are embeddability into $S Sigma_g$, and that this can be improved to $4 g+4$ under the concept of an arrangements used by Ortner.

Embeddability of Arrangements of Pseudocircles and Graphs on Surfaces

- MathematicsDiscret. Comput. Geom.
- 2020

An arrangement of pseudocircles is embeddable into an orientable surface of genus g if and only if all of its subarrangements of size at most $$4g+4$$ 4 g + 4 are.

Selected Open and Solved Problems in Computational Synthetic Geometry

- Mathematics
- 2016

Computational Synthetic Geometry was the title of the Springer Lecture Notes of the first author with Bernd Sturmfels in 1989. During the last 25 years combinatorial structures such as abstract…

Interview with Tomaž Pisanski Toufik Mansour

- Education
- 2021

Photo by Mojca Petrič Tomaž Pisanski studied at the University of Ljubljana where he obtained a B.Sc., M.Sc., and Ph.D. in mathematics. He completed his Ph.D. thesis under the guidance of Torrence…

## References

SHOWING 1-10 OF 36 REFERENCES

Oriented matroids and complete-graph embeddings on surfaces

- MathematicsJ. Comb. Theory, Ser. A
- 2007

Computational Synthetic Geometry

- Mathematics
- 1989

Computational synthetic geometry aims to develop algorithms to find for a given abstract geometric object either a coordinatization over some field or a proof that such a realization does not exist.…

Connected ( n k ) configurations exist for almost all n

- Mathematics
- 2021

A geometric (nk) configuration is a collection of points and straight lines, typically in the Euclidean plane, so that each line passes through k of the points and each of the points lies on k of the…

Quasi-configurations: building blocks for point-line configurations

- MathematicsArs Math. Contemp.
- 2016

The motivation is the problem of the existence of $(n_4)$ configurations, still open for few remaining values of $n, which is based on quasi-configurations: point-line incidence structures where each point is incident to at least $3$ lines and each line is incidentto at least$3$ points.

Enumerating topological $(n_k)$-configurations

- MathematicsCCCG 2011
- 2011

This work provides an algorithm for generating, for given $n and $k$, all topological $(n_k)$-configurations up to combinatorial isomorphism, without enumerating first all combinatorsial $(n-k) $- configurations.

Graphs on Surfaces

- MathematicsJohns Hopkins series in the mathematical sciences
- 2001

This chapter discusses Embeddings Combinatorially, Contractibility, of Cycles, and the Genus Problem, which focuses on planar graphs and the Jordan Curve Theorem, and colorings of Graphs on Surfaces, which are 5-choosable.

Drawing plane graphs nicely

- Computer ScienceActa Informatica
- 2004

Two efficient algorithms for drawing plane graphs nicely draw all edges of a graph as straight line segments without crossing lines if possible, in a way that every inner face and the complement of the outer face are convex polygons.