- The_rigidity_of_infinite_graphs
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**The rigidity of infinite graphs.** / Kitson, D.; C. Power, S.

Research output: Contribution to journal › Journal article › peer-review

Kitson, D & C. Power, S 2018, 'The rigidity of infinite graphs', *Discrete and Computational Geometry*, vol. 60, no. 3, pp. 531-557. https://doi.org/10.1007/s00454-018-9993-0

Kitson, D., & C. Power, S. (2018). The rigidity of infinite graphs. *Discrete and Computational Geometry*, *60*(3), 531-557. https://doi.org/10.1007/s00454-018-9993-0

Kitson D, C. Power S. The rigidity of infinite graphs. Discrete and Computational Geometry. 2018 Oct;60(3):531-557. https://doi.org/10.1007/s00454-018-9993-0

@article{745a38d3b1d54754b86bad7c18e19bd2,

title = "The rigidity of infinite graphs",

abstract = "A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces $(\bR^d,\|\cdot \|_q)$, for $d\geq 2$ and $1 combinatorial characterisation of generic infinitesimal rigidity for finite graphs in $(\bR^2,\|\cdot \|_2)$. Also Tay's multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in $(\bR^d,\|\cdot\|_2)$ is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit $G= \varinjlim G_k$ of an inclusion tower of finite graphs $G_1 \subseteq G_2 \subseteq \dots$ for which the inclusions satisfy a relative rigidity property. For $d\geq 3$ a countable graph which is rigid for generic placements in $\bR^d$ may fail the stronger property of sequential rigidity, while for $d=2$ the properties are equivalent. ",

keywords = "math.MG, math.CO, 52C25, 05C63",

author = "D. Kitson and {C. Power}, S.",

note = "The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0",

year = "2018",

month = oct,

doi = "10.1007/s00454-018-9993-0",

language = "English",

volume = "60",

pages = "531--557",

journal = "Discrete and Computational Geometry",

issn = "0179-5376",

publisher = "Springer New York",

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TY - JOUR

T1 - The rigidity of infinite graphs

AU - Kitson, D.

AU - C. Power, S.

N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0

PY - 2018/10

Y1 - 2018/10

N2 - A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces $(\bR^d,\|\cdot \|_q)$, for $d\geq 2$ and $1 combinatorial characterisation of generic infinitesimal rigidity for finite graphs in $(\bR^2,\|\cdot \|_2)$. Also Tay's multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in $(\bR^d,\|\cdot\|_2)$ is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit $G= \varinjlim G_k$ of an inclusion tower of finite graphs $G_1 \subseteq G_2 \subseteq \dots$ for which the inclusions satisfy a relative rigidity property. For $d\geq 3$ a countable graph which is rigid for generic placements in $\bR^d$ may fail the stronger property of sequential rigidity, while for $d=2$ the properties are equivalent.

AB - A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces $(\bR^d,\|\cdot \|_q)$, for $d\geq 2$ and $1 combinatorial characterisation of generic infinitesimal rigidity for finite graphs in $(\bR^2,\|\cdot \|_2)$. Also Tay's multi-graph characterisation of generic infinitesimal rigidity for finite body-bar frameworks in $(\bR^d,\|\cdot\|_2)$ is generalised to the non-Euclidean norms and to countably infinite graphs. For all dimensions and norms it is shown that a generically rigid countable simple graph is the direct limit $G= \varinjlim G_k$ of an inclusion tower of finite graphs $G_1 \subseteq G_2 \subseteq \dots$ for which the inclusions satisfy a relative rigidity property. For $d\geq 3$ a countable graph which is rigid for generic placements in $\bR^d$ may fail the stronger property of sequential rigidity, while for $d=2$ the properties are equivalent.

KW - math.MG

KW - math.CO

KW - 52C25, 05C63

U2 - 10.1007/s00454-018-9993-0

DO - 10.1007/s00454-018-9993-0

M3 - Journal article

VL - 60

SP - 531

EP - 557

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -