2020
DOI: 10.48550/arxiv.2005.12568
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Topological Drawings meet Classical Theorems from Convex Geometry

Abstract: In this article we discuss classical theorems from Convex Geometry in the context of topological drawings. In a simple topological drawing of the complete graph Kn, any two edges share at most one point: either a common vertex or a point where they cross. Triangles of simple topological drawings can be viewed as convex sets, this gives a link to convex geometry.We present a generalization of Kirchberger's Theorem, a family of simple topological drawings with arbitrarily large Helly number, and a new proof of a… Show more

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(2 citation statements)
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“…3 We imagine the vertices on a horizontal line, and thus if x < y then we may say that x is to the left from y and so on. 4 Let F denote the family of the complements of the hyperedges of F. It is easy to see and was shown in [18] that if F is ABA-free then F is also ABA-free. 5 Notice that the top-and bottomvertices depend only on F and not on H itself.…”
Section: Basic Definitionsmentioning
confidence: 99%
See 1 more Smart Citation
“…3 We imagine the vertices on a horizontal line, and thus if x < y then we may say that x is to the left from y and so on. 4 Let F denote the family of the complements of the hyperedges of F. It is easy to see and was shown in [18] that if F is ABA-free then F is also ABA-free. 5 Notice that the top-and bottomvertices depend only on F and not on H itself.…”
Section: Basic Definitionsmentioning
confidence: 99%
“…Definition 5. A hypergraph H on an ordered vertex set is a pseudohalfplane hypergraph if there exists an ABA-free F on the same ordered vertex set 4 such that H ⊆ F ∪ F. Call T = H ∩ F the topsets and B = H ∩ F the bottomsets, observe that both T and B are ABA-free. The unskippable vertices of F (resp.…”
Section: Introductionmentioning
confidence: 99%