TY - JOUR

T1 - An inductive construction of minimally rigid body-hinge simple graphs

AU - Kobayashi, Yuki

AU - Higashikawa, Yuya

AU - Katoh, Naoki

AU - Kamiyama, Naoyuki

N1 - Publisher Copyright:
© 2014 Elsevier B.V.

PY - 2014

Y1 - 2014

N2 - A d-dimensional body-hinge framework is a collection of d-dimensional rigid bodies connected by hinges, where a hinge is a (d- 2)-dimensional affine subspace, i.e., pin-joints in 2-space, line-hinges in 3-space, plane-hinges in 4-space and etc. Bodies are allowed to move continuously in Rd so that the relative motion of any two bodies connected by a hinge is a rotation around it and the framework is called rigid if every motion provides a framework isometric to the original one. A body-hinge framework is expressed as a pair (G, p) of a multigraph G = (V, E) and a mapping p from e∈. E to a (d- 2)-dimensional affine subspace p(e) in Rd. Namely, v∈V corresponds to a body and uv∈E corresponds to a hinge p(uv) which joins the two bodies corresponding to u and v. Then, G is said to be realized as a body-hinge framework (G, p) in Rd, and is called a body-hinge graph. It is known [9,12] that the infinitesimal rigidity of a generic body-hinge framework (G, p) is determined only by its underlying graph G. So, a graph G is called (minimally) rigid if G can be realized as a (minimally) infinitesimally rigid body-hinge framework in d-dimension. In this paper, we shall present an inductive construction for minimally rigid body-hinge simple graphs in d-dimension with d≥ 3.

AB - A d-dimensional body-hinge framework is a collection of d-dimensional rigid bodies connected by hinges, where a hinge is a (d- 2)-dimensional affine subspace, i.e., pin-joints in 2-space, line-hinges in 3-space, plane-hinges in 4-space and etc. Bodies are allowed to move continuously in Rd so that the relative motion of any two bodies connected by a hinge is a rotation around it and the framework is called rigid if every motion provides a framework isometric to the original one. A body-hinge framework is expressed as a pair (G, p) of a multigraph G = (V, E) and a mapping p from e∈. E to a (d- 2)-dimensional affine subspace p(e) in Rd. Namely, v∈V corresponds to a body and uv∈E corresponds to a hinge p(uv) which joins the two bodies corresponding to u and v. Then, G is said to be realized as a body-hinge framework (G, p) in Rd, and is called a body-hinge graph. It is known [9,12] that the infinitesimal rigidity of a generic body-hinge framework (G, p) is determined only by its underlying graph G. So, a graph G is called (minimally) rigid if G can be realized as a (minimally) infinitesimally rigid body-hinge framework in d-dimension. In this paper, we shall present an inductive construction for minimally rigid body-hinge simple graphs in d-dimension with d≥ 3.

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U2 - 10.1016/j.tcs.2014.08.007

DO - 10.1016/j.tcs.2014.08.007

M3 - Article

AN - SCOPUS:84925114554

VL - 556

SP - 2

EP - 12

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

IS - C

ER -