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Constructing isostatic frameworks for the ℓ1and ℓ-plane

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Published
Article numberP2.49
<mark>Journal publication date</mark>12/06/2020
<mark>Journal</mark>The Electronic Journal of Combinatorics
Issue number2
Volume27
Number of pages21
Publication StatusPublished
<mark>Original language</mark>English

Abstract

We use a new coloured multi-graph constructive method to prove that if the edge-set of a graph G=(V,E) has a partition into two spanning trees T1 and T2
then there is a map p:V→R2, p(v)=(p1(v),p2(v)), such that |pi(u)−pi(v)|⩾|p3−i(u)−p3−i(v)| for every edge uv in Ti(i=1,2). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the ℓ1
or ℓ-plane. We also show how to adapt this technique to incorporate symmetry and indicate several related open problems on rigidity, redundant rigidity and forced symmetric rigidity in normed spaces.