Hierarchical Decompositions of dihypergraphs

TL;DR

This paper introduces a polynomial-time method for hierarchically decomposing dihypergraphs into simpler components using an $ ext{H}$-tree, with applications to closure systems and lattices.

Contribution

It presents a novel polynomial-time algorithm for hierarchical dihypergraph decomposition and introduces H-factors for partially decomposable dihypergraphs.

Findings

Polynomial-time algorithm for dihypergraph decomposition

Introduction of H-factors for indecomposable dihypergraphs

Applications to closure systems and lattices

Abstract

In this paper we are interested in decomposing a dihypergraph $\mathcal{H} = (V, \mathcal{E})$ into simpler dihypergraphs, that can be handled more efficiently. We study the properties of dihypergraphs that can be hierarchically decomposed into trivial dihypergraphs, \ie vertex hypergraph. The hierarchical decomposition is represented by a full labelled binary tree called $\mathcal{H}$-tree, in the fashion of hierarchical clustering. We present a polynomial time and space algorithm to achieve such a decomposition by producing its corresponding $\mathcal{H}$-tree. However, there are dihypergraphs that cannot be completely decomposed into trivial components. Therefore, we relax this requirement to more indecomposable dihypergraphs called H-factors, and discuss applications of this decomposition to closure systems and lattices.