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On the uniqueness of collections of pennies and marbles

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On the uniqueness of collections of pennies and marbles. / Dewar, Sean; Grasegger, Georg; Kubjas, Kaie et al.
In: Examples and Counterexamples, Vol. 7, 100181, 30.06.2025.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Dewar, S, Grasegger, G, Kubjas, K, Mohammadi, F & Nixon, A 2025, 'On the uniqueness of collections of pennies and marbles', Examples and Counterexamples, vol. 7, 100181. https://doi.org/10.1016/j.exco.2025.100181

APA

Dewar, S., Grasegger, G., Kubjas, K., Mohammadi, F., & Nixon, A. (2025). On the uniqueness of collections of pennies and marbles. Examples and Counterexamples, 7, Article 100181. Advance online publication. https://doi.org/10.1016/j.exco.2025.100181

Vancouver

Dewar S, Grasegger G, Kubjas K, Mohammadi F, Nixon A. On the uniqueness of collections of pennies and marbles. Examples and Counterexamples. 2025 Jun 30;7:100181. Epub 2025 Feb 21. doi: 10.1016/j.exco.2025.100181

Author

Dewar, Sean ; Grasegger, Georg ; Kubjas, Kaie et al. / On the uniqueness of collections of pennies and marbles. In: Examples and Counterexamples. 2025 ; Vol. 7.

Bibtex

@article{e05bdc8829614c4ab429c6905522691c,
title = "On the uniqueness of collections of pennies and marbles",
abstract = "In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit d-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.",
keywords = "Globally rigid, Penny graph, Rigid, Unit sphere graph",
author = "Sean Dewar and Georg Grasegger and Kaie Kubjas and Fatemeh Mohammadi and Anthony Nixon",
year = "2025",
month = feb,
day = "21",
doi = "10.1016/j.exco.2025.100181",
language = "English",
volume = "7",
journal = "Examples and Counterexamples",
issn = "2666-657X",
publisher = "Elsevier BV",

}

RIS

TY - JOUR

T1 - On the uniqueness of collections of pennies and marbles

AU - Dewar, Sean

AU - Grasegger, Georg

AU - Kubjas, Kaie

AU - Mohammadi, Fatemeh

AU - Nixon, Anthony

PY - 2025/2/21

Y1 - 2025/2/21

N2 - In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit d-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.

AB - In this note we study the uniqueness problem for collections of pennies and marbles. More generally, consider a collection of unit d-spheres that may touch but not overlap. Given the existence of such a collection, one may analyse the contact graph of the collection. In particular we consider the uniqueness of the collection arising from the contact graph. Using the language of graph rigidity theory, we prove a precise characterisation of uniqueness (global rigidity) in dimensions 2 and 3 when the contact graph is additionally chordal. We then illustrate a wide range of examples in these cases. That is, we illustrate collections of marbles and pennies that can be perturbed continuously (flexible), are locally unique (rigid) and are unique (globally rigid). We also contrast these examples with the usual generic setting of graph rigidity.

KW - Globally rigid

KW - Penny graph

KW - Rigid

KW - Unit sphere graph

U2 - 10.1016/j.exco.2025.100181

DO - 10.1016/j.exco.2025.100181

M3 - Journal article

VL - 7

JO - Examples and Counterexamples

JF - Examples and Counterexamples

SN - 2666-657X

M1 - 100181

ER -