The Excluded Minors for Isometric Realizability in the Plane

Fiorini, Samuel; Huynh, Tony; Joret, Gwenaël; Varvitsiotis, Antonios

doi:10.1137/16m1064775

Cited by 9 publications

(6 citation statements)

References 18 publications

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“…If Lemma 3.5(1) holds then we have a contradiction since adding back v and its 3 neighbours would violate (2, 2, 0)-gain-sparsity. So Lemma 3.5 (2) holds and H bc ∪ H bd is balanced with f (H bc ∪ H bd ) = 2. However by gain switching we can make all edges incident to v have gain 1 and hence H bc ∪ h bd ∪ v is balanced, again contradicting (2, 2, 0)-gain-sparsity.…”

Section: The Gain Of a Path Of Directed Edgesmentioning

confidence: 90%

“…(See [18,19,20] e.g.). This transfer of knowledge between fundamental and applied researchers is one of the motivations for exploring constraint systems in new geometric contexts, such as the normed spaces considered in this article (see also [1,2] for related problems). Another strong motivation comes from the potential for developing combinatorial Laman-type characterisations ( [8]) of rigid graphs in any dimension, due to the amenability of the matroidal sparsity counts arising in some of these contexts.…”

Section: Introductionmentioning

confidence: 99%

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Rigidity of symmetric frameworks in normed spaces

Kitson¹,

Nixon²,

Schulze³

2018

Preprint

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We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwelltype sparsity counts are identified for a large class of d-dimensional normed spaces (including all p spaces with p = 2). Complete combinatorial characterisations are obtained for half-turn rotation in the 1 and ∞ -plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of (2, 2, 0)-gain-tight graphs. Contents 1. Introduction. 1 2. Symmetric frameworks and gain sparsity 2 3. An inductive construction of (2, 2, 0)-gain tight graphs 11 4. Application to C 2 -symmetric frameworks in the 1 and ∞ -plane 21 5. Concluding remarks 30 References 30

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Section: The Gain Of a Path Of Directed Edgesmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

Rigidity of symmetric frameworks in normed spaces

Kitson¹,

Nixon²,

Schulze³

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…Lemma 24 ( [12]). Let G be a 2-connected graph with distinct vertices u and v such that deg G (w) ≥ 3 for all w ∈ V (G) \{u, v}.…”

Section: -Connected Graphsmentioning

confidence: 99%

“…Therefore, for all p ∈ [1, ∞], K 3 is the only excluded minor for f p (G) ≤ 1. Fiorini, Huynh, Joret, and Varvitsiotis [12] determined that W 4 , the wheel on 5 vertices, and the graph K 4 + e K 4 (see Figure 1) are the only excluded minors for f ∞ (G) ≤ 2 and for f 1 (G) ≤ 2. As far as we know, the complete set of excluded minors for f p (G) ≤ k is unknown for all other values of p and k.…”

Section: Introductionmentioning

confidence: 99%

Unavoidable minors for graphs with large $\ell_p$-dimension

Fiorini¹,

Huynh²,

Joret³

et al. 2019

Preprint

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A metric graph is a pair (G, d), where G is a graph and d :for all edges vw ∈ E(G). The p-dimension of G is the least integer k such that there exists an isometric embedding of (G, d) in k p for all distance functions d such that (G, d) has an isometric embedding in K p for some K. It is easy to show that p-dimension is a minor-monotone property. In this paper, we characterize the minor-closed graph classes C with bounded p-dimension, for p ∈ {2, ∞}. For p = 2, we give a simple proof that C has bounded 2-dimension if and only if C has bounded treewidth. In this sense, the 2-dimension of a graph is 'tied' to its treewidth.For p = ∞, the situation is completely different. Our main result states that a minorclosed class C has bounded ∞-dimension if and only if C excludes a graph obtained by joining copies of K4 using the 2-sum operation, or excludes a Möbius ladder with one 'horizontal edge' removed.

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“…The study of the rigidity of frameworks in normed spaces can be seen to date back to Kitson and Power [23], while research into flexible motions of frameworks in normed spaces was considered earlier by Cook, Lovett and Morgan [12]. The closely related topic of graph flattenability -determining whether all placements of a graph in an infinite-dimensional ℓ p space can be embedded into a d-dimensional ℓ p space whilst preserving edge lengths -dates back even further to the work of Holsztynski [20], Witsenhausen [30], and Ball [4], and has been further continued in recent years by Willoughby and Sitharam [29] and Fiorini, Huynh, Joret, and Varvitsiotis [17].…”

mentioning

confidence: 99%

Infinitesimal rigidity and prestress stability for frameworks in normed spaces

Dewar¹

2021

Preprint

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A (bar-and-joint) framework is a set of points in a normed space with a set of fixed distance constraints between them. Determining whether a framework is locally rigid -i.e. whether every other suitably close framework with the same distance constraints is an isometric copy -is NP-hard when the normed space has dimension 2 or greater. We can reduce the complexity by instead considering derivatives of the constraints, which linearises the problem. By applying methods from non-smooth analysis, we shall strengthen previous sufficient conditions for framework rigidity that utilise first-order derivatives. We shall also introduce the notions of prestress stability and second-order rigidity to the topic of normed space rigidity, two weaker sufficient conditions for framework rigidity previously only considered for Euclidean spaces.

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The Excluded Minors for Isometric Realizability in the Plane

Abstract: Let G be a graph and p ∈ [1, ∞]. The parameter f p (G) is the least integer k such that for all m and all vectors (r

Cited by 9 publications

References 18 publications

Rigidity of symmetric frameworks in normed spaces

Rigidity of symmetric frameworks in normed spaces

Unavoidable minors for graphs with large $\ell_p$-dimension

Infinitesimal rigidity and prestress stability for frameworks in normed spaces

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