Faces of Directed Edge Polytopes

TL;DR

This paper characterizes the facets of directed edge polytopes derived from finite quivers, extending previous results on symmetric edge polytopes to directed cases, and describes faces of all dimensions when a rank function exists.

Contribution

It provides a complete characterization of facets and faces of directed edge polytopes for arbitrary finite quivers, generalizing known results to directed graphs.

Findings

Facets characterized by connectivity and rank function existence.

Extension of symmetric edge polytope results to directed edge polytopes.

Complete description of faces when a rank function is present.

Abstract

Given a finite quiver (directed graph) without loops and multiedges, the convex hull of the column vector of the incidence matrix is called the directed edge polytope and is an interesting example of lattice polytopes. In this paper, we give a complete characterization of facets of the directed edge polytope of an arbitrary finite quiver without loops and multiedges in terms of the connectivity and the existence of a rank function. Our result can be regarded as an extension of the result of Higashitani et al. on facets of symmetric edge polytopes to directed edge polytopes. When the quiver in question has a rank function, we obtain a characterization of faces of arbitrary dimensions.