On Flattenability of Graphs

We consider a generalization of the concept of d-flattenability of graphs introduced for the l2 norm by Belk and Connelly to general lp norms, with integer P , 1 ≤ p < ∞, though many of our results work for l∞ as well. The following results are shown for graphs G, using notions of genericity, rigidity, and generic d-dimensional rigidity matroid introduced by Kitson for frameworks in general lp norms, as well as the cones of vectors of pairwise l p distances of a finite point configuration in d-dimensional, lp space: (i) d-flattenability of a graph G is equivalent to the convexity of d-dimensional, inherent Cayley configurations spaces for G, a concept introduced by the first author; (ii) d-flattenability and convexity of Cayley configuration spaces over specified non-edges of a d-dimensional framework are not generic properties of frameworks (in arbitrary dimension); (iii) d-flattenability of G is equivalent to all of G’s generic frameworks being d-flattenable; (iv) existence of one generic dflattenable framework for G is equivalent to the independence of the edges of G, a generic property of frameworks; (v) the rank of G equals the dimension of the projection of the d-dimensional stratum of the l p distance cone. We give stronger results for specific norms for d = 2: we show that (vi) 2-flattenable graphs for the l1-norm (and l∞-norm) are a larger class than 2-flattenable graphs for Euclidean l2-norm case and finally (vii) prove further results towards characterizing 2-flattenability in the l1-norm. A number of conjectures and open problems are posed.