Characterizing redundant rigidity and redundant global rigidity of body-hinge graphs

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Highlights

  • We characterize the redundant rigidity and the redundant global rigidity of body-hinge graphs in Rd in terms of graph connectivity.

  • Our result implies that both edge-redundancy for rigidity and edge-redundancy for global rigidity can be checked via efficient graph-connectivity algorithms.

  • 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) to a 2-edge-rigid bar-joint graph with a minimum number of added edges is NP-hard.

Abstract

In this paper, we characterize the redundant rigidity and the redundant global rigidity of body-hinge graphs in Rd in terms of graph connectivity.

Although an efficient algorithm which determines mixed-connectivity is still not known, our result implies that both edge-redundancy for rigidity and edge-redundancy for global rigidity can be checked via efficient graph-connectivity algorithms.

Introduction

The aim of this paper is to characterize the redundant rigidity and the redundant global rigidity of body-hinge graphs in Rd 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 k1 and h1, respectively. A graph G is (k,h)-connected if removing any (k1) 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 Rd); 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 G=(V,E), where V and E represent a set of bodies and a set of hinges, respectively. Namely uvE corresponds to a hinge p(uv) (i.e. a (d2)-dimensional affine subspace) which joins the two bodies u and v. G is said to be realized as a body-hinge framework (G,p) in Rd, and is called a body-hinge graph. When a body-hinge graph G can be realized as an infinitesimally rigid body-hinge framework in Rd, 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 Rd with d2 if and only if (D1)G contains D edge-disjoint spanning trees, where D=(d+12) and (D1)G denotes the graph obtained from G by replacing each edge by (D1) parallel edges.

In the following, a graph G is called h-edge-rigid in Rd if removing any (h1) edges from G results in a graph which is rigid in Rd. The reader should keep in mind that rigidity (also h-edge rigidity and (k,h)-rigidity; see below) of a graph is ambiguous unless the underlying dimension is specified. Our definitions and results apply to any dimension d (d2); 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 k1 and h1, respectively. A graph G is called (k,h)-rigid in Rd with d2 if removing any (k1) vertices from G results in a graph which is h-edge-rigid in Rd.

Furthermore, our work has applications to global rigidity. We say that (G,p) is globally rigid in Rd if every d-dimensional framework which is equivalent to (G,p) is congruent to (G,p) (see [4] for details). A graph G is globally rigid in Rd if every (or equivalently, if some) generic realization of G in Rd is globally rigid. A graph G is called h-edge-globally rigid in Rd if removing any (h1) edges from G results in a graph which is globally rigid in Rd.

We now define the notion of redundant global rigidity for graphs.

Definition 3 Redundant global rigidity

Let k and h be integers such that k1 and h1, respectively. A graph G is called (k,h)-globally rigid in Rd with d2 if removing any (k1) vertices from G results in a graph which is h-edge-globally rigid in Rd.

The main result of this paper is stated in the following theorem.

Theorem 1

Let k and h be integers such that k1 and h2, respectively.

(1) A graph G is (k,h)-rigid in R2 if and only if G is (k,h+1)-connected and G is (k,h)-globally rigid in R2 if and only if G is (k,h+2)-connected.

(2) For any d3, the following three statements are equivalent for any graph G: (i) G is (k,h)-rigid in Rd, (ii) G is (k,h)-globally rigid in Rd, (iii) G is (k,h+1)-connected.

Section snippets

Preliminaries

White and Whiteley [19] defined the infinitesimal motions of a body-hinge framework by using real vectors of length (d+12), called screw centers. (G,p) is said to be infinitesimally rigid if all infinitesimal motions of (G,p) are trivial (see [5] for details). Tay [17] and Whiteley [20] independently proved that the infinitesimal rigidity of a generic body-hinge framework (G,p) 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 d2, G is (k,h)-rigid in Rd if and only if G is (k,h+1)-connected. (b) G is (k,h)-globally rigid in R2 if and only if G is (k,h+2)-connected. (c) For any d3, G is (k,h)-rigid in Rd if and only if G is (k,h)-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 (k,h)-rigid in Rd if G is not (k,h+1)

Conclusion

We characterized the redundant rigidity and the redundant global rigidity of graphs in Rd 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.

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    Supported by JSPS Grant-in-Aid for Scientific Research (A) (25240004).

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