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• The_rigidity_of_infinite_graphs

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

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## The rigidity of infinite graphs

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### Standard

In: Discrete and Computational Geometry, Vol. 60, No. 3, 10.2018, p. 531-557.

Research output: Contribution to journalJournal articlepeer-review

### Harvard

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

### Vancouver

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

### Author

Kitson, D. ; C. Power, S. / The rigidity of infinite graphs. In: Discrete and Computational Geometry. 2018 ; Vol. 60, No. 3. pp. 531-557.

### Bibtex

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$the properties are equivalent. ", keywords = "math.MG, math.CO, 52C25, 05C63", author = "D. Kitson and {C. Power}, S.", note = "The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0", year = "2018", month = oct, doi = "10.1007/s00454-018-9993-0", language = "English", volume = "60", pages = "531--557", journal = "Discrete and Computational Geometry", issn = "0179-5376", publisher = "Springer New York", number = "3", } ### RIS TY - JOUR T1 - The rigidity of infinite graphs AU - Kitson, D. AU - C. Power, S. N1 - The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0 PY - 2018/10 Y1 - 2018/10 N2 - A rigidity theory is developed for the Euclidean and non-Euclidean placements of countably infinite simple graphs in the normed spaces$(\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 -