$\renewcommand{\epsilon}{\varepsilon}$
Consider the convex polyhedron $P$ determined by the following inequalities:
\begin{gather*}
0\leq x\leq\alpha, \quad 0\leq y\leq\beta, \quad 0\leq z\leq\gamma,\\
\delta_1\leq x+y\leq\delta_2, \quad \epsilon_1\leq x-y\leq\epsilon_2,\\
\zeta_1\leq x+z\leq\zeta_2, \quad \eta_1\leq x-z\leq\eta_2,\\
\kappa_1\leq y+z\leq\kappa_2, \quad \lambda_1\leq y-z\leq\lambda_2,
\end{gather*}
where we'll say for now that $\alpha=\beta=\gamma=10,~$ $\delta_1=\zeta_1=\kappa_1=1,~$ $\delta_2=\zeta_2=\kappa_2=19,~$ $\epsilon_1=\eta_1=\lambda_1=-9,~$ and $\epsilon_2=\eta_2=\lambda_2=9$. I will refer to these Greek letters as parameters. With these values the polyhedron is called a chamfered cube; a picture is shown below.
Consider the graph $\Gamma_P$ determined by the edges and vertices of $P$. It is clear that if we change any of the parameters by a tiny amount—say 0.1 with the current values—the resulting polyhedron will have the same graph. However, we can of course change the values so much that the graph changes.
My question is, what is the set of values of the parameters such that the graph is the same as in the picture? What methods would you use to go about finding such a set of relations? I guess I'm trying to determine some sort of moduli space. Any help at all would be appreciated, even just a literature reference.
You just would have to calculate the sizes of the edges as functions of your parameters. Then you can vary those parameters freely, as long as all those edge sizes would remain positive.
--- rk