Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

Jackson, Bill; Jordán, Tibor; Szabadka, Zoltan

doi:10.1007/s00454-005-1225-8

Cited by 49 publications

(65 citation statements)

References 18 publications

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“…We note that for the plane Jackson, Jordán and Szabadka have an alternative proof that edge-splitting preserves global rigidity [14]. This proof has recently been generalized to all dimensions.…”

Section: Prior Results On Global Rigidity and Infinitesimal Rigiditymentioning

confidence: 84%

“…Given the characterization of globally rigid graphs in the plane, the methods have recently been extended to characterize globally linked pairs of vertices in some classes of graphs in R 2 [14]. These are pairs of vertices whose distance is the same in all frameworks which are equivalent to any given generic framework of the graph.…”

Section: Globally Linked Pairsmentioning

confidence: 99%

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Generic global rigidity of body–bar frameworks

Connelly

Jordán

Whiteley

2013

Journal of Combinatorial Theory, Series B

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A basic geometric question is to determine when a given framework G(p) is globally rigid in Euclidean space R d , where G is a finite graph and p is a configuration of points corresponding to the vertices of G. G(p) is globally rigid in R d if for any other configuration q for G such that the edge lengths of G(q) are the same as the corresponding edge lengths of G(p), then p is congruent to q. A framework G(p) is redundantly rigid, if it is rigid and it remains rigid after the removal of any edge of G. Hendrickson [9] proved that redundant rigidity is a necessary condition for generic global rigidity, as is (d+1)-connectivity.Recent results in [2] and [10] have shown that when the configuration p is generic and d = 2, redundant rigidity and 3-connectivity are also sufficient -a good combinatorial characterization that only depends on G and can be checked in polynomial time. It appears that a similar result for d = 3 is beyond the scope of present techniques However, there is a special class of generic frameworks that have polynomial time algorithms for their generic rigidity (and redundant rigidity) in R d for any d ≥ 1, as shown in [19], namely generic bodyand-bar frameworks. Such frameworks are constructed from a finite number of rigid bodies that are connected by bars generically placed with respect to each body. We show that a body-and-bar framework is generically globally rigid in R d , for any d ≥ 1, if it is redundantly rigid. As a consequence there is a deterministic polynomial time combinatorial algorithm on the graph to determine the generic global rigidity of body-and-bar frameworks in any dimension.

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Section: Prior Results On Global Rigidity and Infinitesimal Rigiditymentioning

confidence: 84%

Section: Globally Linked Pairsmentioning

confidence: 99%

Generic global rigidity of body–bar frameworks

Connelly

Jordán

Whiteley

2013

Journal of Combinatorial Theory, Series B

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“…A bar framework G(p) is said to be redundantly rigid if G(p) remains rigid after deleting any one edge of G. Recently, the problem of global rigidity of bar frameworks has received a great deal of attention [28,41,61,62]. Hendrickson [58,59] proved that if a generic framework G(p) in R r with at least r + 1 vertices is globally rigid, then the graph G = (V, E) is r + 1 vertex-connected and G(p) is redundantly rigid.…”

Section: Bar Framework Global Rigiditymentioning

confidence: 99%

Euclidean distance matrices, semidefinite programming and sensor network localization

Alfakih¹,

Piccialli²,

Wolkowicz³

2011

Port. Math.

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The fundamental problem of distance geometry, F P DG , involves the characterization and study of sets of points based only on given values of (some of) the distances between pairs of points. This problem has a wide range of applications in various areas of mathematics, physics, chemistry, and engineering. Euclidean Distance Matrices, EDM , play an important role in F P DG . They use the squared distances and provide elegant and powerful convex relaxations for F P DG . These EDM problems are closely related to graph realization, GRL ; and graph rigidity, GRD , plays an important role. Moreover, by relaxing the embedding dimension restriction, EDM problems *

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“…[32] Let (G, p) be a generic framework, x, y ∈ V (G), xy / ∈ E(G), and suppose that κ G (x, y) ≤ 2. Then {x, y} is not globally linked in (G, p).…”

Section: Globally Linked Pairs and Uniquely Localizable Nodesmentioning

confidence: 99%

Graph Theoretic Techniques in the Analysis of Uniquely Localizable Sensor Networks

Jackson

Jordán

2009

Localization Algorithms and Strategies for Wireless Sensor Networks

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In the network localization problem the goal is to determine the location of all nodes by using only partial information on the pairwise distances (and by computing the exact location of some nodes, called anchors). The network is said to be uniquely localizable if there is a unique set of locations consistent with the given data. Recent results from graph theory and combinatorial rigidity made it possible to characterize uniquely localizable networks in two dimensions. Based on these developments, extensions, related optimization problems, algorithms, and constructions also became tractable. This chapter gives a detailed survey of these new results from the graph theorist's viewpoint.

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Globally Linked Pairs of Vertices in Equivalent Realizations of Graphs

Cited by 49 publications

References 18 publications

Generic global rigidity of body–bar frameworks

Generic global rigidity of body–bar frameworks

Euclidean distance matrices, semidefinite programming and sensor network localization

Graph Theoretic Techniques in the Analysis of Uniquely Localizable Sensor Networks

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