Indecomposability and beyond via the graph of edge dependencies

TL;DR

The paper introduces a unifying framework for certifying indecomposability of polytopes via the graph of edge dependencies, augmented by implicit edges, degenerate edges, and covering flats to bound deformation cones. It develops a main indecomposability theorem and a suite of tools (projections, subframeworks, and implicit edges) that subsume prior criteria and yield new indecomposable deformed permutahedra, notably P_{n,m} and Q_{n,m} from truncated graphical zonotopes of complete bipartite graphs. These constructions produce infinite families of indecomposable deformed permutahedra that are not matroid polytopes and provide new rays for the submodular cone, while also enabling refined deformation-cone bounds and product-like behavior. The results extend to deep truncations, affine projections, and parallelogramic Minkowski sums, including stacking operations, thereby broadening the toolkit for deformation-cone analysis and polytope rigidity in geometric combinatorics.

Abstract

A polytope is called indecomposable if it cannot be expressed (non-trivially) as a Minkowski sum of other polytopes. Since the concept was introduced by Gale in 1954, several increasingly strong criteria have been developed to characterize indecomposability. Our first contribution is a new indecomposability criterion that unifies and generalizes most of the previous techniques. The key new ingredient of our method is the introduction of the graph of (implicit) edge dependencies, which has broader applications in the study of deformation cones of polytopes, beyond indecomposability. One of our main applications is providing new indecomposable deformed permutahedra that are not matroid polytopes. In 1970, Edmonds posed the problem of characterizing the extreme rays of the submodular cone, that is, indecomposable deformed permutahedra. Matroid polytopes from connected matroids give one such family of polytopes. We provide a new infinite disjoint family by taking certain graphical zonotopes and deeply truncating 1 or 2 specific vertices. In this way, we construct $2 \lfloor\frac{n-1}{2}\rfloor$ new indecomposable deformations of the $n$-permutahedron in $\mathbb{R}^n$. We also showcase other applications of our tools. For example, we use them to refute a conjecture by Smilansky (1987) stating that an indecomposable polytope needs to have few vertices with respect to its number of facets. We provide bounds on the dimension of deformation cones and characterize certain of their rays, we introduce parallelogramic Minkowski sums whose deformation cone can be written as a product of deformation cones, and we construct indecomposable polytopes via truncations and stackings.

Figures (15)

• Figure 1: (Left) The regular hexagon can be written as the Minkowski sum of indecomposable polytopes in several ways. (Right) Two deformations of the regular hexagon, where each edge $\mathsf{e}$ is labeled by the coordinate of the associated edge-length vector $\dv(\mathsf{Q})_{\mathsf{e}}$.
• Figure 2: (Left to right) The $3$-cube, a prism over a triangle, a hemicube, a pyramid on a hexagon. Each polytope is in blue. In red, green and cyan, the subgraphs of $ED(\mathsf{P})$ obtained by linking opposite edges of parallelograms and edges of triangles. In each case, the graphs $ED(\mathsf{P})$ is the (union of) cliques on the subgraphs drawn. Note that the $3$-cube is the Minkowski sum of 3 linearly independent segments; the prism over a triangle is the sum of a triangle and a (linearly independent) segment; the hemicube is the sum of 2 orthogonal triangles. The pyramid is indecomposable.
• Figure 3: Kallay1982 (left) A decomposable realization of a twice-stacked cube. (Right) An indecomposable realization of a twice-stacked cube, where the edges $AB$ and $XY$ are not co-planar. In each, we draw some edges of $ED(\mathcal{F})$: edges of the same color are in a common clique of $ED(\mathcal{F})$. On the right, using the rigid cycle (whose convex hull is a tetrahedron) we prove that $AB$ and $XY$ are also part of the red component.
• Figure 4: (Left) Projecting the framework on top, parallel to the plane spanned by its blue edges, give rise to a segment with 2 degenerated edges. Consequently, the 2 red edges are dependent because they are projected to the same edge (while the blue edges are not dependent). (Right) Projecting the framework on top, parallel to the direction spanned by its blue edges, give rise to a triangle with 3 degenerated edges and degenerated blue vertices. Consequently, the 3 red edges are dependent (while the blue edges are not).
• Figure 5: (Left) A triangular cupola $J_{3}$. (Middle) A 3-dimensional polytope constructed as the sum of 2 triangles which share 1 edge direction. (Right) A chiseled cube, obtained by lowering the $z$-coordinate of two diagonally opposite vertices. Their deformation cones have dimension 2, and have 2 rays.

Theorems & Definitions (180)

• Theorem A: \ref{['thm:mainthm']}
• Theorem B: \ref{['cor:BetterLowerBound']}
• Lemma 2.1.1
• proof
• Lemma 2.1.2
• proof
• Remark 2.1.3
• Lemma 2.1.4: Shephard1963, Kallay1982
• Remark 2.1.5
• Definition 2.2.1