Root polytopes and Jaeger-type dissections for directed graphs

TL;DR

This paper introduces a novel approach to associating root polytopes with directed graphs, especially semi-balanced ones, using ribbon structures to achieve shellable dissections that facilitate the computation of new graph invariants.

Contribution

It develops a method to dissect root polytopes of directed graphs into simplices, enabling the calculation of the $h^*$-vector and revealing properties like product formulas and connections to greedoids.

Findings

Shellable dissections of root polytopes for semi-balanced graphs

Computation of the $h^*$-vector and its properties

Recovery of known triangulations for layer-complete graphs

Abstract

We associate root polytopes to directed graphs and study them by using ribbon structures. Most attention is paid to what we call the semi-balanced case, i.e., when each cycle has the same number of edges pointing in the two directions. Given a ribbon structure, we identify a natural class of spanning trees and show that, in the semi-balanced case, they induce a shellable dissection of the root polytope into maximal simplices. This allows for a computation of the $h^*$-vector of the polytope and for showing some properties of this new graph invariant, such as a product formula and that in the planar case, the $h^*$-vector is equivalent to the greedoid polynomial of the dual graph. We obtain a general recursion relation as well. We also work out the case of layer-complete directed graphs, where our method recovers a previously known triangulation. Indeed our dissection is often but not always a triangulation; we address this with a series of examples.