Confluence of the Node-Domination and Edge-Domination Hypergraph Rewrite Rules
Antoine Amarilli, Mikaël Monet, Rémi De Pretto
TL;DR
The paper investigates two natural hypergraph rewrite rules—edge-domination and node-domination—and proves they are confluent up to isomorphism. By leveraging termination and Newman's lemma, it reduces the problem to local confluence and carries out a comprehensive case analysis of rule interactions, including mixed applications. The authors lift the rewriting process to hypergraph isomorphism classes and show that any two reduction paths from a given hypergraph can be continued to yield isomorphic results, implying a unique minimal hypergraph up to isomorphism. This formalizes a robust, order-insensitive preprocessing step for problems like computing minimum hitting sets, and reinforces the structural predictability of hypergraph simplifications. The work thus provides a solid theoretical foundation for using edge- and node-domination in hypergraph simplification and related transversal studies.
Abstract
In this note, we study two rewrite rules on hypergraphs, called edge-domination and node-domination, and show that they are confluent. These rules are rather natural and commonly used before computing the minimum hitting sets of a hypergraph. Intuitively, edge-domination allows us to remove hyperedges that are supersets of another hyperedge, and node-domination allows us to remove nodes whose incident hyperedges are a subset of that of another node. We show that these rules are confluent up to isomorphism, i.e., if we apply any sequences of edge-domination and node-domination rules, then the resulting hypergraphs can be made isomorphic via more rule applications. This in particular implies the existence of a unique minimal hypergraph, up to isomorphism.