Abstract
The notion of vertex separability by partial edges for a simple hypergraph is introduced and the related structural properties of the hypergraph are analyzed in terms of maximal (with respect to set-theoretic inclusion) compacts and of dividers, where a compact is a vertex set in which every two vertices are separated by no partial edge, and a divider is a partial edge X for which there exists a pair of vertices that are separated by X and by no proper subset of X. It is proven that, given a hypergraph H, the hypergraph (called the compaction of H) made up of maximal compacts of H is acyclic and coincides with H if and only if H is acyclic; furthermore, it has the same dividers as H, and can be characterized as being the unique acyclic hypergraph that has the same compacts as H. Polynomial algorithms for finding maximal compacts and dividers of a given hypergraph are provided. Finally, an application to the problem of computing the maximum-entropy extension of a system of marginals over a hypergraph is discussed.
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