A Structural Decomposition for HypergraphsPETER JEAVONS ,

Hypergraphs arise in a variety of applications and are commonly classiied as cyclic or acyclic. In this paper we develop a more reened classiication scheme for cyclic hypergraphs based on a natural decomposition strategy. The fundamental building blocks in our decompositions are subsets of edges known as k-hinges. For any hypergraph, a set of more than k of its edges is deened to be a k-hinge if all connected components of the hypergraph with respect to the set of edges meet the latter within at most k of its edges. A k-hinge tree is a set of minimal k-hinges that cover all edges of H , and form a tree with respect to intersection. The size of the largest node in any 1-hinge tree is shown to be an invariant of the hypergraph, which we call the degree of cyclicity. Acyclic hypergraphs are hypergraphs with degree of cyclicity 2. The concept of degree of cyclicity was rst presented by Gyssens in the context of re-lational database design, but is presented here for arbitrary hypergraphs with a greatly simpliied proof. For more general k-hinges we show that it is possible to obtain more powerful decompositions. However, in this case there may be several possible decompositions which do not share any structural invariant. We therefore consider restrictions on k-hinges which are necessary in order to guarantee this structural invariant.