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Modulus of hypertrees

Huy Truong, Pietro Poggi-Corradini

TL;DR

This work extends the discrete modulus framework from graphs and matroids to hypergraphs by introducing hypertrees and the multi-tree family, and by establishing a precise Fulkerson duality between partition inequalities and hypertree multisets. It shows that 1-modulus of hypertrees coincides with the (weighted) strength of the underlying hypergraphic matroid, and that 2-modulus reveals a hierarchical decomposition of hypergraphs via a Hypergraph Decomposition Process that identifies homogeneous cores and partitions whose shrunk subhypergraphs are partition-connected. The results unify strength, fractional arboricity, and modulus in both hypertree and multi-tree settings, and they extend classical dualities to a hypergraph context. The framework yields a principled way to quantify the richness of hypertree families, elucidate hierarchical structure in arbitrary hypergraphs, and generalize prior dualities from graphs and matroids to hypergraphic matroids.

Abstract

Lorea [11] and later Frank et al. [8] generalized graphic matroids to hypergraphic matroids. In [8], the authors introduced hypertrees as a generalization of spanning trees and proved a form of the theorem of Tutte [18] and Nash-Williams [14]. In [3, 15, 17], the authors explored the modulus of the family of spanning trees in graphs and of the family of bases of matroids, and provided connections to the notions of strength and fractional arboricity. They also established Fulkerson duality for these families. In this paper, we extend these results to hypertrees, and show that the modulus of hypertrees uncovers a hierarchical structure within arbitrary hypergraphs.

Modulus of hypertrees

TL;DR

This work extends the discrete modulus framework from graphs and matroids to hypergraphs by introducing hypertrees and the multi-tree family, and by establishing a precise Fulkerson duality between partition inequalities and hypertree multisets. It shows that 1-modulus of hypertrees coincides with the (weighted) strength of the underlying hypergraphic matroid, and that 2-modulus reveals a hierarchical decomposition of hypergraphs via a Hypergraph Decomposition Process that identifies homogeneous cores and partitions whose shrunk subhypergraphs are partition-connected. The results unify strength, fractional arboricity, and modulus in both hypertree and multi-tree settings, and they extend classical dualities to a hypergraph context. The framework yields a principled way to quantify the richness of hypertree families, elucidate hierarchical structure in arbitrary hypergraphs, and generalize prior dualities from graphs and matroids to hypergraphic matroids.

Abstract

Lorea [11] and later Frank et al. [8] generalized graphic matroids to hypergraphic matroids. In [8], the authors introduced hypertrees as a generalization of spanning trees and proved a form of the theorem of Tutte [18] and Nash-Williams [14]. In [3, 15, 17], the authors explored the modulus of the family of spanning trees in graphs and of the family of bases of matroids, and provided connections to the notions of strength and fractional arboricity. They also established Fulkerson duality for these families. In this paper, we extend these results to hypertrees, and show that the modulus of hypertrees uncovers a hierarchical structure within arbitrary hypergraphs.

Paper Structure

This paper contains 9 sections, 20 theorems, 104 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypergraphs and hypertrees
  4. Discrete modulus
  5. Fulkerson duality for hypertrees
  6. Modulus of the hypertree family
  7. Modulus for the multi-tree family
  8. Symmetry property for modulus
  9. Modulus for the multi-tree family

Key Result

Theorem 1.1

Given a hypergraph $H = (V,E)$, let $M(H)$ be the associated hypergraphic matroid of $H$. For $F \subseteq E$, the rank $r(F)$ of $F$ is given by the following formula:

Figures (2)

  • Figure 1: A hypergraph $H =(V,E)$ with $V =\left\{ v_1,v_2,v_3 \right\} ,$$E = \left\{ e_1,e_2\right\}$, and $e_1 = e_2 =\left\{ v_1,v_2,v_3 \right\}$.
  • Figure 2: A hypergraph where hyperedges are styled according to the optimal density $\eta^*$. Solid edges are present with $\eta^* = \frac{3}{5}$, dashed edges with $\eta^* = 1$ and dotted edges with $\eta^* = \frac{3}{2}$.

Theorems & Definitions (54)

  • Theorem 1.1: Frank
  • Theorem 1.2: Frank
  • Theorem 1.3: Frank
  • Definition 1.4
  • Example 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 44 more