Deformation cones of graphical zonotopes for $K_4$-free graph

TL;DR

This work analyzes deformation cones of graphical zonotopes associated to $K_4$-free graphs, proving a precise triangulation of $\mathbb{DC}(\mathsf{Z}_{G})$ into $2^{|T|}$ simplicial cones with rays given by $\mathsf{\Delta}_e$ for edges and $\pm\mathsf{\Delta}_t$ for triangles, yielding a clear description of 2D and 3D faces and their Minkowski summands. The main result shows a remarkably constrained structure: for $K_4$-free graphs the deformation cone is governed by low-dimensional rays, and this triangulation fails in graphs containing $K_4$, where higher-dimensional indecomposable summands can arise. The paper also explores geometric realizations such as the hexa-rhombic dodecahedron in the bi-triangle graph, provides a thorough account of 2D deformations (parallelograms and hexagons), and presents computational evidence of high-dimensional phenomena beyond the $K_4$-free regime. Open questions address not-$K_4$-free cases, complete $f$-vector characterizations, and broader classes of generalized permutahedra with low Minkowski dimension. These findings deepen the understanding of the submodular cone’s faces via graphical zonotopes and offer a framework for analyzing Minkowski summands in related polyhedral settings.

Abstract

In this paper, we compute a triangulation of certain faces of the submodular cone. More precisely, graphical zonotopes are generalized permutahedra, and hence their deformation cones are faces of the submodular cone. We give a triangulation of these faces for graphs without induced complete sub-graph on 4 vertices. We deduce the rays of these faces: Minkowski indecomposable deformations of these graphical zonotopes are segments and triangles. Besides, computer experiments lead to examples of graphs without induced complete sub-graph on 5 vertices, whose graphical zonotopes have high dimensional Minkowski indecomposable deformations.

Figures (6)

• Figure 1: Graphical zonotopes of dimension 2 and 3. We only need to consider connected graphs on 4 nodes or less, as, for any non-connected graph $G$, the graphical zonotope $\ZG$ is normally equivalent to the one of any 1-sum of the connected components of $G$.
• Figure 2: Each vertex and edge is labeled by the corresponding ordered partition: acyclic orientations of $G$ for vertices, and acyclic orientations of a contraction of $\contrG[x]$ for a pair of parallel edges.
• Figure 3: (PPP2023gZono). For $K_3$, the graphical zonotope $\ZG[K_3]$ is a regular hexagon, i.e. the 2-dimensional permutahedron (bottom left). Its deformation cone $\mathbb{DC}(\mathsf{Z}_{K_3})$ is 4-dimensional. We picture a 3-dimensional affine section of $\mathbb{DC}(\mathsf{Z}_{K_3})$. The deformations of $\ZG[K_3]$ corresponding to some of the points of $\mathbb{DC}(\mathsf{Z}_{K_3})$ are depicted. Especially, all points in the interior correspond to polytopes normally equivalent to $\ZG[K_3]$, while the above left polytope is Loday's associahedron.
• Figure 4: The polytopes associated to the 2-dimensional faces of $\mathbb{DC}(\mathsf{Z}_{G})$ for $G$ a $K_4$-free graph.
• Figure 5: (Left) Hexa-rhombic dodecahedron. (Right) The graph of the hexa-rhombic dodecahedron, obtained as a Schlegel projection on one of its hexagonal facets. Edges with the same color have the same length, and the two labeled hexagons are parallel in the hexa-rhombic dodecahedron. Note that there are other edges with the same length, but we do not use them.

Theorems & Definitions (30)

• Theorem A: \ref{['thm:DCZG_K4-free_graphs', 'cor:DCZG_graph_is_bi-pyramid_graph']}
• Definition 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• proof
• Remark 2.5
• Definition 2.6
• Proposition 2.7
• Remark 2.8