Deformed graphical zonotopes

TL;DR

This paper characterizes the deformation cone of graphical zonotopes, generalizing known results about permutahedra, and identifies conditions under which this cone is simplicial based on graph properties.

Contribution

It provides an explicit, irredundant description of the deformation cone of graphical zonotopes, linking geometric deformations to graph-theoretic features.

Findings

Deformation cone described by independent equations and facet inequalities.

Faces of the simplex form a basis for the deformation cone.

Deformation cone is simplicial if and only if the graph is triangle-free.

Abstract

We study deformations of graphical zonotopes. Deformations of the classical permutahedron (which is the graphical zonotope of the complete graph) have been intensively studied in recent years under the name of generalized permutahedra. We provide an irredundant description of the deformation cone of the graphical zonotope associated to a graph $G$, consisting of independent equations defining its linear span (in terms of non-cliques of $G$) and of the inequalities defining its facets (in terms of common neighbors of neighbors in $G$). In particular, we deduce that the faces of the standard simplex corresponding to induced cliques in $G$ form a linear basis of the deformation cone, and that the deformation cone is simplicial if and only if $G$ is triangle-free.