On Flattenability of Graphs

TL;DR

This paper generalizes the concept of graph flattenability from Euclidean to general $l_p$ norms, establishing key equivalences with convexity of configuration spaces and properties of frameworks, with specific results for $l_1$ and $l_$ norms.

Contribution

It extends flattenability theory to $l_p$ norms, linking it to convexity, rigidity, and independence properties, and provides new characterizations for 2-flattenability in $l_1$ and $l_$ norms.

Findings

Flattenability is equivalent to convexity of Cayley configuration spaces.

Flattenability and convexity are not generic properties in arbitrary dimensions.

Existence of a generic flattenable framework implies edge independence.

Abstract

We consider a generalization of the concept of $d$-flattenability of graphs - introduced for the $l_2$ norm by Belk and Connelly - to general $l_p$ norms, with integer $P$, $1 \le p < \infty$, though many of our results work for $l_\infty$ 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 $l_p$ norms, as well as the cones of vectors of pairwise $l_p^p$ distances of a finite point configuration in $d$-dimensional, $l_p$ 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 $d$-flattenable 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 $l_p^p$ distance cone. We give stronger results for specific norms for $d = 2$: we show that (vi) 2-flattenable graphs for the $l_1$-norm (and $l_\infty$-norm) are a larger class than 2-flattenable graphs for Euclidean $l_2$-norm case and finally (vii) prove further results towards characterizing 2-flattenability in the $l_1$-norm. A number of conjectures and open problems are posed.