Bergeacyclic

Bergeacyclic is a property of hypergraphs in graph theory. A hypergraph H = (V, E) is Bergeacyclic if it contains no Berge cycle. A Berge cycle of length k is a sequence of distinct vertices v1, v2, ..., vk and distinct hyperedges e1, e2, ..., ek such that vi and vi+1 both belong to ei for each i (with vk+1 interpreted as v1). In other words, the cycle alternates between vertices and hyperedges in which each consecutive vertex lies in the corresponding hyperedge.

The notion, named after Claude Berge, generalizes the idea of a cycle from ordinary graphs to hypergraphs. Bergeacyclic hypergraphs are one of several acyclicity notions used in hypergraph theory, alongside alpha-acyclicity, beta-acyclicity, and gamma-acyclicity. Bergeacyclicity is a relatively broad form of acyclicity, and checking it is important in certain algorithmic contexts.

Examples include ordinary trees viewed as hypergraphs with edges that do not create Berge cycles, and more generally any hypergraph that can be built without forming a Berge cycle. Conversely, a hypergraph containing a Berge cycle is not Bergeacyclic.

Applications of Bergeacyclicity appear in database theory, constraint satisfaction, and logic, where acyclic structures often enable more efficient query evaluation and reasoning. It provides a framework for understanding how higher-arity relations interact without generating cyclical dependencies.

See also: Berge cycle, hypergraph, incidence graph, alpha-acyclicity, beta-acyclicity, gamma-acyclicity.

References: Claude Berge, Hypergraphs, and surveys on acyclicity notions in hypergraph theory.