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A directed edge polytope $\mathcal{A}_G$ is a lattice polytope arising fromroot 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 andreflexive, that is, one can associate this polytope to a Gorenstein toric Fanovariety $X_G$ with terminal singularities. It is shown by Totaro that a toricFano variety which is smooth in codimension $2$ and $\mathbb{Q}$-factorial incodimension $3$ 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 $3$.

Rigid Gorenstein toric Fano varieties arising from directed graphs