Decomposability of abstract and path-induced convexities in hypergraphs

An abstract convexity space on a connected hypergraph H with vertex set V (H) is a family C of subsets of V (H) (to be called the convex sets of H) such that: (i) C contains the empty set and V (H), (ii) C is closed under intersection, and (iii) every set in C is connected in H. A convex set X of H is a minimal vertex convex separator of H if there exist two vertices of H that are separated by X and are not separated by any convex set that is a proper subset of X. A nonempty subset X of V (H) is a cluster of H if in H every two vertices in X are not separated by any convex set. The cluster hypergraph of H is the hypergraph with vertex set V (H) whose edges are the maximal clusters of H. A convexity space on H is called decomposable if it satisfies the following three properties: (C1) the cluster hypergraph of H is acyclic, (C2) every edge of the cluster hypergraph of H is convex, (C3) for every nonempty proper subset X of V (H), a vertex v does not belong to the convex hull of X if and only if v is separated from X in H by a convex cluster. It is known that the monophonic convexity (i.e., the convexity induced by the set of chordless paths) on a connected hypergraph is decomposable. In this paper we first provide two characterizations of decomposable convexities and then, after introducing the notion of a hereditary path family in a connected hypergraph H, we show that the convexity space on H induced 494 F.M. Malvestuto and M. Moscarini by any hereditary path family containing all chordless paths (such as the families of simple paths and of all paths) is decomposable.