## Characterizing minimally flat symmetric hypergraphs

### Abstract

In [Kaszanitzky, Schulze, Lifting symmetric pictures to polyhedral scenes, EGRES TR-2015-07] we gave necessary conditions for a symmetric $d$-picture (i.e., a symmetric realization of an incidence structure in $\mathbb{R}^d$) to be minimally flat, that is, to be non-liftable to a polyhedral scene without having redundant constraints. These conditions imply very simply stated restrictions on the number of those structural components of the picture that are fixed by the elements of its symmetry group. In this paper we show that these conditions on the fixed structural components, together with the standard non-symmetric counts, are also sufficient for a plane picture which is generic with three-fold rotational symmetry $\mathcal{C}_3$ to be minimally flat. This combinatorial characterization of minimally flat $\mathcal{C}_3$-generic pictures is obtained via a new inductive construction scheme for symmetric sparse hypergraphs. We also give a sufficient condition for sharpness of pictures with $\mathcal{C}_3$ symmetry.

Bibtex entry:

@techreport{egres-15-17,
AUTHOR = {Kaszanitzky, Vikt{\'o}ria and Schulze, Bernd},
TITLE = {Characterizing minimally flat symmetric hypergraphs},
NOTE= {{\tt www.cs.elte.hu/egres}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2015},
NUMBER = {TR-2015-17}
}