# Almost all string graphs are intersection graphs of plane convex sets

Pach J, Reed B, Yuditsky Y. 2020. Almost all string graphs are intersection graphs of plane convex sets. Discrete and Computational Geometry. 63(4), 888–917.

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Author

Pach, János

^{IST Austria}; Reed, Bruce; Yuditsky, YelenaDepartment

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Abstract

A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.

Publishing Year

Date Published

2020-06-05

Journal Title

Discrete and Computational Geometry

Volume

63

Issue

4

Page

888-917

ISSN

eISSN

IST-REx-ID

### Cite this

Pach J, Reed B, Yuditsky Y. Almost all string graphs are intersection graphs of plane convex sets.

*Discrete and Computational Geometry*. 2020;63(4):888-917. doi:10.1007/s00454-020-00213-zPach, J., Reed, B., & Yuditsky, Y. (2020). Almost all string graphs are intersection graphs of plane convex sets.

*Discrete and Computational Geometry*. Springer Nature. https://doi.org/10.1007/s00454-020-00213-zPach, János, Bruce Reed, and Yelena Yuditsky. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.”

*Discrete and Computational Geometry*. Springer Nature, 2020. https://doi.org/10.1007/s00454-020-00213-z.J. Pach, B. Reed, and Y. Yuditsky, “Almost all string graphs are intersection graphs of plane convex sets,”

*Discrete and Computational Geometry*, vol. 63, no. 4. Springer Nature, pp. 888–917, 2020.Pach, János, et al. “Almost All String Graphs Are Intersection Graphs of Plane Convex Sets.”

*Discrete and Computational Geometry*, vol. 63, no. 4, Springer Nature, 2020, pp. 888–917, doi:10.1007/s00454-020-00213-z.**All files available under the following license(s):**

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arXiv 1803.06710