Canonical and monophonic convexities in hypergraphs

Known properties of ‘‘canonical connections’’ from database theory and of ‘‘closed sets’’ from statistics implicitly define a hypergraph convexity, here called canonical convexity (cconvexity), and provide an efficient algorithm to compute c-convex hulls. We characterize the class of hypergraphs in which c-convexity enjoys the Minkowski–Krein–Milman property. Moreover, we compare c-convexity with the natural extension to hypergraphs of monophonic convexity (orm-convexity), and prove that: (1)m-convexity is coarser than cconvexity, (2) m-convexity and c-convexity are equivalent in conformal hypergraphs, and (3)m-convex hulls can be computed in the same efficient way as c-convex hulls. © 2009 Elsevier B.V. All rights reserved.