Graph minors, Ehrhart theory, and a monotonicity property

TL;DR

This paper investigates the behavior of the $h^*$-polynomial of extended root polytopes associated with directed graphs, demonstrating monotonicity under edge deletion and contraction, and identifying cases where the polynomial remains unchanged.

Contribution

It establishes a monotonicity property of the $h^*$-polynomial under graph operations and characterizes cases with invariant polynomials, linking graph operations to Ehrhart theory.

Findings

$h^*$-polynomial coefficients do not increase under edge deletion and contraction.

Identifies conditions when the $h^*$-polynomial remains unchanged, such as contracting edges in a minimal directed join.

Connects graph minors with Ehrhart theory and Gorenstein properties of lattice polytopes.

Abstract

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 $h^*$-polynomial of the associated extended root polytope increase. We examine cases when the $h^*$-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.