Realizable Dimension of Periodic Frameworks

Belk and Connelly introduced the realizable dimension $\text{rd}(G)$ of a finite graph $G$, which is the minimum nonnegative integer $d$ such that every framework $(G,p)$ in any dimension admits a framework in ${\mathbb{R}}^d$ with the same edge lengths. They characterized finite graphs with realizable dimension at most $1$, $2$, or $3$ in terms of forbidden minors. In this paper, we consider periodic frameworks and extend the notion to $\mathbb{Z}$-symmetric graphs. We give a forbidden minor characterization of $\mathbb{Z}$-symmetric graphs with realizable dimension at most $1$ or $2$, and show that the characterization can be checked in polynomial time for given quotient $\mathbb{Z}$-labelled graphs.