Confluence of the Node-Domination and Edge-Domination Hypergraph Rewrite Rules

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.