Rights statement: The final publication is available at Springer via http://dx.doi.org/10.1007/s00454-018-9993-0
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Final published version
Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
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 -