Intersecting Diametral Balls Induced by a Geometric Graph

Olimjoni, Pirahmad,; Vasilevskii, A. G.

doi:10.1007/s00454-022-00457-x

Cited by 3 publications

(3 citation statements)

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Section: Similarly a Graph Is An Open Tverberg Graph If The Open Ball...mentioning

confidence: 99%

“…In [6], the authors established that for any even set of distinct points in R 𝑑 , there exists a perfect matching that is an open Tverberg graph. They also questioned whether a max-sum matching satisfies this property, as discussed in [6,Problem 9.3]. While Theorem 2 answers this question affirmatively for 𝑑 = 2, our approach relies heavily on the arrangement of points in the plane and cannot be extended to higher dimensions.…”

Section: Similarly a Graph Is An Open Tverberg Graph If The Open Ball...mentioning

confidence: 99%

“…In 1966, Helge Tverberg [8] proved that for any (𝑟 − 1)(𝑑 + 1) + 1 points in R 𝑑 , there exists a partition of them into 𝑟 parts whose convex hulls intersect. This paper studies a variation of Tverberg's theorem recently introduced in [5,7], and further develops ideas presented in [2,6].…”

Section: Introductionmentioning

confidence: 96%

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Intersecting balls induced by a geometric graph II

Polina¹

2023

Preprint

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For a graph whose vertices are points in R 𝑑 , consider the closed balls with diameters induced by its edges. The graph is called a Tverberg graph if these closed balls intersect.A max-sum tree of a finite point set 𝑋 ⊂ R 𝑑 is a tree with vertex set 𝑋 that maximizes the sum of Euclidean distances of its edges among all trees with vertex set 𝑋. Similarly, a max-sum matching of an even set 𝑋 ⊂ R 𝑑 is a perfect matching of 𝑋 maximizing the sum of Euclidean distances between the matched points among all perfect matchings of 𝑋.We prove that a max-sum tree of any finite point set in R 𝑑 is a Tverberg graph, which generalizes a recent result of Abu-Affash et al., who established this claim in the plane. Additionally, we provide a new proof of a theorem by Bereg et al., which states that a max-sum matching of any even point set in the plane is a Tverberg graph. Moreover, we proved a slightly stronger version of this theorem.Theorem 1. A max-sum tree of any finite point set in R 𝑑 is a Tverberg graph.

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Section: Similarly a Graph Is An Open Tverberg Graph If The Open Ball...mentioning

confidence: 99%

Section: Similarly a Graph Is An Open Tverberg Graph If The Open Ball...mentioning

confidence: 99%