Almost all string graphs are intersection graphs of plane convex sets

Pach, János and Reed, B. and Yuditsky, Y. (2018) Almost all string graphs are intersection graphs of plane convex sets. In: 34th International Symposium on Computational Geometry, SoCG 2018. Leibniz International Proceedings in Informatics, LIPIcs (99). Schloss Dagstuhl Leibniz-Zentrum für Informatik, Dagstuhl, pp. 681-6814. ISBN 9783959770668

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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 α→ ∞). 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. © János Pach, Bruce Reed, and Yelena Yuditsky; licensed under Creative Commons License CC-BY 34th Symposium on Computational Geometry (SoCG 2018).

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