Graph minors, Ehrhart theory, and a monotonicity property

We study the extended root polytope associated to a directed graph. We show that under the operations of deletion and contraction of an edge of the graph, none of the coefficients of the $h^{\ast }$-polynomial of the associated extended root polytope increase. We examine cases when the $h^{\ast }$-polynomial does not change, for instance when contracting the edges of a minimal directed join in a digraph whose lattice polytope has the Gorenstein property.