Global rigidity of triangulations with braces

Jordán, Tibor; Tanigawa, Shin‐ichi

doi:10.1016/j.jctb.2018.11.003

Cited by 18 publications

(18 citation statements)

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“…Observe that in this unibraced triangulation the three vertices that are not in the brace must span a nonfacial triangle of P . The following is implicit in [11]. Also see [7] for related results.…”

Section: Unibraced Triangulationsmentioning

confidence: 96%

“…In other words, every unibraced triangulation can be constructed from this single irreducible by a sequence of vertex splitting moves of a specific kind. Triangulated spheres with a single brace have previously been studied in [20] in relation to redundant rigidity and more recently in [7,11] in relation to global rigidity. The case b = 2 is more involved and in Section 4 we show that there are exactly five distinct irreducibles (see Figures 3,4,5,6,7).…”

Section: Introductionmentioning

confidence: 99%

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Braced triangulations and rigidity

Cruickshank¹,

Kastis²,

Kitson³

et al. 2021

Preprint

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We consider the problem of finding an inductive construction, based on vertex splitting, of triangulated spheres with a fixed number of additional edges (braces). We show that for any positive integer b there is such an inductive construction of triangulations with b braces, having finitely many base graphs. In particular we establish a bound for the maximum size of a base graph with b braces that is linear in b. In the case that b = 1 or 2 we determine the list of base graphs explicitly. Using these results we show that doubly braced triangulations are (generically) minimally rigid in two distinct geometric contexts arising from a hypercylinder in R 4 and a class of mixed norms on R 3 . 2020 Mathematics Subject Classification. 52C25, 05C10, 46B20. E.K. and D.K.

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Section: Unibraced Triangulationsmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Braced triangulations and rigidity

Cruickshank¹,

Kastis²,

Kitson³

et al. 2021

Preprint

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“…Theorem 2.4 above). It is conjectured in [15] that globally rigid graphs in R d are "non-degenerate" (for the definition see [15]). Since non-degenerate graphs are M-connected ([15, Lemma 3.2]), Theorem 3.5 gives an affirmative answer to a weaker, M-connected version of this conjecture.…”

Section: Globally Rigid Graphs Are M-connectedmentioning

confidence: 99%

Globally rigid graphs are fully reconstructible

Garamvölgyi¹,

Gortler²,

Jordán³

2021

Preprint

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A d-dimensional framework is a pair (G, p), where G = (V, E) is a graph and p is a map from V to R d . The length of an edge uv ∈ E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in R d if the graph G and its edge lengths uniquely determine (G, p), up to congruence. A graph G is called globally rigid in R d if every d-dimensional generic framework (G, p) is globally rigid.In this paper, we consider the problem of reconstructing a graph from the set of edge lengths arising from a generic framework. Roughly speaking, a graph G is strongly reconstructible in C d if it is uniquely determined by the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of its vertices. It is known that if G is globally rigid in R d on at least d + 2 vertices, then it is strongly reconstructible in C d . We strengthen this result and show that under the same conditions, G is in fact fully reconstructible in C d , which means that the set of edge lengths alone is sufficient to uniquely reconstruct G, without any constraint on the number of vertices.We also prove that if G is globally rigid in R d on at least d + 2 vertices, then the d-dimensional generic rigidity matroid of G is connected. This result generalizes Hendrickson's necessary condition for global rigidity and gives rise to a new combinatorial necessary condition.Finally, we provide new families of fully reconstructible graphs and use them to answer some questions regarding unlabeled reconstructibility posed in recent papers.

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“…A recent result of Jordán and Tanigawa [17] characterises when graphs constructed from plane triangulations by adding some additional edges are globally rigid in R 3 .…”

Section: Introductionmentioning

confidence: 99%