Rigid Gorenstein toric Fano varieties arising from directed graphs

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 \(3\).