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Hypergraph Acyclicity Revisited

The notion of graph acyclicity has been extended to several different notions of hypergraph acyclicity, in increasing order of generality: gamma acyclicity, beta acyclicity, and alpha acyclicity, that have met a great interest in many fields. We prove the equivalence between the numerous characterizations of each notion with a new, simpler proof, in a self-contained manner. For that purpose, we introduce new notions of alpha, beta and gamma leaf that allow to define new "rule-based" characterizations of each notion. The combined presentation of the notions is completed with a study of their respective closure properties. New closure results are established, and alpha, beta and gamma acyclicity are proved optimal w.r.t. their closure properties.