  A directed edge polytope $\mathcal{A}_G$ is a lattice polytope arising from
root system $A_n$ and a finite directed graph $G$. If every directed edge of
$G$ belongs to a directed cycle in $G$, then $\mathcal{A}_G$ is terminal and
reflexive, that is, one can associate this polytope to a Gorenstein toric Fano
variety $X_G$ with terminal singularities. It is shown by Totaro that a toric
Fano variety which is smooth in codimension $2$ and $\mathbb{Q}$-factorial in
codimension $3$ is rigid. In the present paper, we classify all directed graphs
$G$ such that $X_G$ is a toric Fano variety which is smooth in codimension $2$
and $\mathbb{Q}$-factorial in codimension $3$.
