Sums of rank-one matrices and ranks of principal submatrices

Fallat, Shaun M.; Tifenbach, Ryan M.

doi:10.1080/03081087.2017.1406446

Cited by 2 publications

(2 citation statements)

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“…The Cauchy-Binet formula can be used to give a quick proof for the case when each A i has rank one, see [5].…”

Section: Consider the Functionmentioning

confidence: 99%

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Minimally globally rigid graphs

Garamvölgyi¹,

Jordán²

2022

Preprint

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A graph G = (V, E) is globally rigid in R d if for any generic placement p : V → R d of the vertices, the edge lengths p(u) − p(v) , uv ∈ E uniquely determine p, up to congruence. In this paper we consider minimally globally rigid graphs, in which the deletion of an arbitrary edge destroys global rigidity. We prove that if2 . This implies that the minimum degree of G is at most 2d + 1. We also show that the only graph in which the upper bound on the number of edges is attained is the complete graph K d+2 . It follows that every minimally globally rigid graph in R d on at least d + 3 vertices is flexible in R d+1 .As a counterpart to our main result on the sparsity of minimally globally rigid graphs, we show that in two dimensions, dense graphs always contain non-trivial globally rigid subgraphs. More precisely, if some graph G = (V, E) satisfies |E| ≥ 5|V |, then G contains a globally rigid subgraph on at least seven vertices in R 2 . If the well-known "sufficient connectivity conjecture" is true, then this result also extends to higher dimensions.Finally, we discuss a conjectured strengthening of our main result, which states that if a pair of vertices {u, v} is linked in G in R d+1 , then {u, v} is globally linked in G in R d . We prove this conjecture in the d = 1, 2 cases, along with a variety of related results.

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“…The Cauchy-Binet formula can be used to give a quick proof for the case when each A i has rank one, see [5].…”

Section: Consider the Functionmentioning

confidence: 99%

“…We do not now whether the constant 5 in Corollary 5.7 can be further improved. See Figure 4 for a family of graphs with approximately 5 2 |V | edges and no non-trivial globally rigid subgraphs in R 2 .…”

Section: Globally Rigid Subgraphs In Dense Graphsmentioning

confidence: 99%