Global Rigidity of Unit Ball Graphs

Garamvölgyi, Dániel; Jordán, Tibor

doi:10.1137/18m1220662

Cited by 2 publications

(2 citation statements)

References 26 publications

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“…If a = b, we find g t (0) > 0 and 2bt > g t (b) = t 2 + s 2 for t sufficiently small. Hence, also in this case (14) has a solution x t 3 with x t 3 = O(t).…”

Section: Right-and Left-setmentioning

confidence: 90%

“…Before moving on, we would like to mention, however, that our setting is closely related to notions in graph-rigidity theory [3,19,22]. In fact, the notation above corresponds to associating to each Z 2 configuration C = {x k } n k=1 ∈ Z 2n the unit-disk graph G = (V, E), namely, a graph with one vertex for each point in C, and with an edge between two vertices whenever the corresponding points in C are at distance 1 [14]. In particular, the set of edges E can be specified as…”

Section: Relation With Rigidity In Graphsmentioning

confidence: 99%

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Stability of Z2 configurations in 3D

2021

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Inspired by the issue of stability of molecular structures, we investigate the strict minimality of point sets with respect to configurational energies featuring two- and three-body contributions. Our main focus is on characterizing those configurations which cannot be deformed without changing distances between first neighbours or angles formed by pairs of first neighbours. Such configurations are called angle-rigid. We tackle this question in the class of finite configurations in Z 2 , seen as planar three-dimensional point sets. A sufficient condition preventing angle-rigidity is presented. This condition is also proved to be necessary when restricted to specific subclasses of configurations.

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“…If a = b, we find g t (0) > 0 and 2bt > g t (b) = t 2 + s 2 for t sufficiently small. Hence, also in this case (14) has a solution x t 3 with x t 3 = O(t).…”

Section: Right-and Left-setmentioning

confidence: 90%

Section: Relation With Rigidity In Graphsmentioning

confidence: 99%

Stability of Z2 configurations in 3D

2021

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Globally Linked Pairs of Vertices in Generic Frameworks

Jordán,

Villányi

2024

Combinatorica

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A d-dimensional framework is a pair (G, p), where $$G=(V,E)$$ G = ( V , E ) is a graph and p is a map from V to $${\mathbb {R}}^d$$ R d . The length of an edge $$xy\in E$$ x y ∈ E in (G, p) is the distance between p(x) and p(y). A vertex pair $$\{u,v\}$$ { u , v } of G is said to be globally linked in (G, p) if the distance between p(u) and p(v) is equal to the distance between q(u) and q(v) for every d-dimensional framework (G, q) in which the corresponding edge lengths are the same as in (G, p). We call (G, p) globally rigid in $${\mathbb {R}}^d$$ R d when each vertex pair of G is globally linked in (G, p). A pair $$\{u,v\}$$ { u , v } of vertices of G is said to be weakly globally linked in G in $${\mathbb {R}}^d$$ R d if there exists a generic framework (G, p) in which $$\{u,v\}$$ { u , v } is globally linked. In this paper we first give a sufficient condition for the weak global linkedness of a vertex pair of a $$(d+1)$$ ( d + 1 ) -connected graph G in $${\mathbb {R}}^d$$ R d and then show that for $$d=2$$ d = 2 it is also necessary. We use this result to obtain a complete characterization of weakly globally linked pairs in graphs in $${\mathbb {R}}^2$$ R 2 , which gives rise to an algorithm for testing weak global linkedness in the plane in $$O(|V|^2)$$ O ( | V | 2 ) time. Our methods lead to a new short proof for the characterization of globally rigid graphs in $${\mathbb {R}}^2$$ R 2 , and further results on weakly globally linked pairs and globally rigid graphs in the plane and in higher dimensions.

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Global Rigidity of Unit Ball Graphs

Cited by 2 publications

References 26 publications

Stability of Z2 configurations in 3D

Stability of Z2 configurations in 3D

Globally Linked Pairs of Vertices in Generic Frameworks

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