Constructing isostatic frameworks for the l(1) and plane l(infinity )plane

摘要

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 T-1 and T-2 then there is a map p : V -> R-2, p(v) = (p(v)(1), p(v)(2)), such that vertical bar p(u)(i) - p(v)(i)vertical bar >= vertical bar p(u)(3-i) - p(v)(3-i)vertical bar for every edge uv in T-i (i = 1, 2). As a consequence, we solve an open problem on the realisability of minimally rigid bar-joint frameworks in the l(1) or l(infinity)-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.