Rigid Gorenstein toric Fano varieties arising from directed graphs

directed edge polytope 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 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 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 ℚ -factorial in codimension 3.