Constructing isostatic frameworks for the l1 and l infinity plane (Pre-published)

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) = (p(v)1,p(v)2), such that |p(u)i −p(v)i| > |p(u)3−i−p(v)3−i| 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 `1 or `∞-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.