Characterizing redundant rigidity and redundant global rigidity of body-hinge graphs
Introduction
The aim of this paper is to characterize the redundant rigidity and the redundant global rigidity of body-hinge graphs in in terms of graph connectivity. Graph connectivity has been extensively studied [1], [13] and several previous studies had investigated the connection between rigidity and graph connectivity in the context of 2-dimensional bar and joint frameworks [3], [8], [15]. The motivation to study body-hinge frameworks is due to their extensive use in real-world applications such as robotics, engineering, material science and computational biology [6], [22]. We now define the notion of mixed-connectivity. Definition 1 Mixed-connectivity Let k and h be integers such that and , respectively. A graph G is -connected if removing any vertices from G results in a graph which is h-edge-connected.
A d-dimensional body-hinge framework is a collection of d-dimensional rigid bodies connected by revolute hinges (see Fig. 1 and [17], [20] for further details). We say a d-dimensional body-hinge framework is rigid if every motion results in a framework isometric to the original one (i.e. the motion corresponds to an isometry of ); such motions are called trivial or rigid-body motions. Otherwise a framework is called flexible [7], [21]. The underlying combinatorial structure of a body-hinge framework is a multigraph , where V and E represent a set of bodies and a set of hinges, respectively. Namely corresponds to a hinge (i.e. a -dimensional affine subspace) which joins the two bodies u and v. G is said to be realized as a body-hinge framework in , and is called a body-hinge graph. When a body-hinge graph G can be realized as an infinitesimally rigid body-hinge framework in , G is called rigid [17], [20]. We call a body-hinge graph simply a graph. Proposition 1 (See [17], [20].) A graph G can be realized as a rigid body-hinge framework in with if and only if contains D edge-disjoint spanning trees, where and denotes the graph obtained from G by replacing each edge by parallel edges.
In the following, a graph G is called h-edge-rigid in if removing any edges from G results in a graph which is rigid in . The reader should keep in mind that rigidity (also h-edge rigidity and -rigidity; see below) of a graph is ambiguous unless the underlying dimension is specified. Our definitions and results apply to any dimension d (); the dimension will be specified in the provided examples.
We now define the notion of redundant rigidity for graphs. Definition 2 Redundant rigidity Let k and h be integers such that and , respectively. A graph G is called -rigid in with if removing any vertices from G results in a graph which is h-edge-rigid in .
Furthermore, our work has applications to global rigidity. We say that is globally rigid in if every d-dimensional framework which is equivalent to is congruent to (see [4] for details). A graph G is globally rigid in if every (or equivalently, if some) generic realization of G in is globally rigid. A graph G is called h-edge-globally rigid in if removing any edges from G results in a graph which is globally rigid in .
We now define the notion of redundant global rigidity for graphs. Definition 3 Redundant global rigidity Let k and h be integers such that and , respectively. A graph G is called -globally rigid in with if removing any vertices from G results in a graph which is h-edge-globally rigid in .
The main result of this paper is stated in the following theorem. Theorem 1 Let k and h be integers such that and , respectively. (1) A graph G is -rigid in if and only if G is -connected and G is -globally rigid in if and only if G is -connected. (2) For any , the following three statements are equivalent for any graph G: (i) G is -rigid in , (ii) G is -globally rigid in , (iii) G is -connected.
Section snippets
Preliminaries
White and Whiteley [19] defined the infinitesimal motions of a body-hinge framework by using real vectors of length , called screw centers. is said to be infinitesimally rigid if all infinitesimal motions of are trivial (see [5] for details). Tay [17] and Whiteley [20] independently proved that the infinitesimal rigidity of a generic body-hinge framework is determined only by its underlying graph G. A body-hinge framework is generic if its rigidity matrix has a maximum
Proof of Theorem 1
In this section, we prove Theorem 1. For this purpose, we prove the following three statements. (a) For , G is -rigid in if and only if G is -connected. (b) G is -globally rigid in if and only if G is -connected. (c) For any , G is -rigid in if and only if G is -globally rigid. When we summarize the above three, we will complete the proof of Theorem 1.
Proof of the only if part in (a) We show the contraposition of the only if part: “G is not -rigid in if G is not
Conclusion
We characterized the redundant rigidity and the redundant global rigidity of graphs in in terms of graph connectivity.
Our result is contrasted with the fact that the problem of augmenting a Laman graph (i.e., the graph corresponding to a minimally rigid generic bar-joint framework in 2-dimension – see [7]) to a 2-edge-rigid bar-joint graph with a minimum number of added edges is NP-hard [2]. By Theorem 1, in order to make any graph G h-edge-rigid in any dimension by adding a minimum number
Acknowledgements
We would like to thank anonymous referees and Shin-ichi Tanigawa for giving a lot of comments on the draft version.
References (22)
- et al.
A sufficient connectivity condition for generic rigidity in the plane
Discrete Appl. Math.
(2009) - et al.
Augmenting edge-connectivity over the entire range in time
Jpn. J. Ind. Appl. Math.
(1999) Connections in Combinatorial Optimization
(2011)- et al.
Augmenting the rigidity of a graph in
Algorithmica
(2011) - et al.
Generic global rigidity of body-hinge frameworks
(2014) - et al.
A proof of the molecular conjecture
Discrete Comput. Geom.
(2011) - et al.
Protein flexibility predictions using graph theory
Proteins
(2001) On graphs and rigidity of plane skeletal structures
J. Eng. Math.
(1970)- et al.
On generic rigidity in the plane
SIAM J. Algebr. Discrete Methods
(1982) - et al.
Certifying 3-edge-connectivity
Computing edge connectivity in multigraphs and capacitated graphs
SIAM J. Discrete Math.
Cited by (2)
Global rigidity
2017, Handbook of Discrete and Computational Geometry, Third EditionRedundancy Optimization of Finite-Dimensional Structures: Concept and Derivative-Free Algorithm
2017, Journal of Structural Engineering (United States)