Constructing isostatic frameworks for the ℓ1and ℓ∞-plane

Clinch, Katie and Kitson, Derek (2020) Constructing isostatic frameworks for the ℓ1and ℓ∞-plane. The Electronic Journal of Combinatorics, 27 (2): P2.49. ISSN 1077-8926

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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.