Extending drawings of complete graphs into arrangements of pseudocircles

Arroyo, Alan and Richter, Bruce and Sunohara, Matthew Extending drawings of complete graphs into arrangements of pseudocircles. (In preparation). (Unpublished)

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Abstract

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 drawing (introduced by Arroyo et al, submitted) has extensions of its edges to simple closed curves such that any two curves intersect in exactly two points. Using this result, we show that} a set of three axioms of simple closed curve extensions characterizes h-convexity.

Item Type: Article
Subjects: 500 Science > 510 Mathematics > 516 Geometry
Research Group: Wagner Group
Depositing User: Alan Arroyo Guevara
Date Deposited: 16 Apr 2019 08:11
Last Modified: 16 Apr 2019 08:11
URI: https://repository.ist.ac.at/id/eprint/1083

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