## Canonical and monophonic convexities in hypergraphs.(English)Zbl 1211.05093

Summary: Known properties of “canonical connections” from database theory and of “closed sets” from statistics implicitly define a hypergraph convexity, here called canonical convexity $$(c$$-convexity), 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 (or $$m$$-convexity), and prove that: (1) $$m$$-convexity is coarser than $$c$$-convexity, (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.

### MSC:

 05C65 Hypergraphs
Full Text:

### References:

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