# Extending drawings of complete graphs into arrangements of pseudocircles

```@article{Arroyo2021ExtendingDO,
title={Extending drawings of complete graphs into arrangements of pseudocircles},
author={Alan Arroyo and R. Richter and M. Sunohara},
journal={SIAM J. Discret. Math.},
year={2021},
volume={35},
pages={1050-1076}
}```
• Published 2021
• Computer Science, Mathematics
• SIAM J. Discret. Math.
We prove three principal results. First we exhibit a drawing of \$K_{10}\$ in the plane for which there do not exist extensions of the edges to simple closed curves with any two curves intersecting at most twice. Second, we exhibit a drawing of \$K_9\$ that has an extension of its edges to simple closed curves such that any two curves intersect in at most two points, but no extension to simple closed curves has every two curves intersecting in exactly two points. Third, we show that every h-convex… Expand
1 Citations

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#### References

SHOWING 1-10 OF 35 REFERENCES
Sweeping arrangements of curves
• Computer Science, Mathematics
• SCG '89
• 1989
A curve can sweep any such arrangement and maintain the k-intersection property if and only if k equals 1 or 2, and is applied to an eclectic set of problems: finding Boolean formulae for polygons with curved edges, counting triangles and digons in arrangements of pseudocircles, and finding extension curves for arrangements. Expand
Extending Drawings of Graphs to Arrangements of Pseudolines
• Computer Science, Mathematics
• SoCG
• 2020
This work extends this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings, and leads to a polynomial-time algorithm to recognize pseudolinar drawings and construct the pseudolines when it is possible. Expand
Arrangements of Pseudocircles: On Circularizability
• Mathematics, Computer Science
• Graph Drawing
• 2018
It is shown that there are exactly four non-circularizable arrangements of 5 pseudocircles, exactly one of them is intersecting and that every digon-free intersecting arrangement of circles contains at least \$2n-4\$ triangles. Expand
Convex Quadrilaterals and k-Sets
• Mathematics
• 2003
We prove that the minimum number of convex quadrilaterals determined by n points in general position in the plane ‐ or in other words, the rectilinear crossing number of the complete graph Kn ‐ is atExpand
Complete Graph Drawings Up to Triangle Mutations
• E. Gioan
• Mathematics, Computer Science
• WG
• 2005
It is proved, constructively, that two complete graph drawings have the same subsketch if and only if they can be transformed into each other by a sequence of triangle mutations – or triangle switches. Expand
All Good Drawings of Small Complete Graphs
Good drawings (also known as simple topological graphs) are drawings of graphs such that any two edges intersect at most once. Such drawings have attracted attention as generalizations of geometricExpand
Levi's Lemma, pseudolinear drawings of Kn, and empty triangles
• Mathematics, Computer Science
• J. Graph Theory
• 2018
A new proof of Levi's Enlargement Lemma for pseudoline arrangements in the real projective plane is proved and proofs that pseudolinear and convex drawings of K_n have \$n^2+{}\$O\$(n\log n)\$ and O\$( n^2)\$, respectively, empty triangles are proved. Expand
Oriented Matroids
• Computer Science, Mathematics
• Handbook of Discrete and Computational Geometry, 2nd Ed.
• 2004
The theory of oriented matroids provides a broad setting in which to model, describe, and analyze combinatorial properties of geometric configurations, among them duality, realizability, the study of simplicial cells, and the treatment of convexity. Expand
Non-Shellable Drawings of Kn with Few Crossings
• Mathematics, Computer Science
• CCCG
• 2014
This paper constructs a non-shellable family of drawings of Kn with exactly H(n) crossings, where every edge in the authors' drawings is intersected by at least one other edge. Expand
On the Number of Crossings in a Complete Graph
• Mathematics
• 1963
The purpose of this article is to describe two problems which involve drawing graphs in the plane. We will discuss both complete graphs and complete bicoloured graphs. The complete graph K n with nExpand