ISRN Combinatorics

Volume 2013 (2013), Article ID 453808, 9 pages

http://dx.doi.org/10.1155/2013/453808

## Decomposable Convexities in Graphs and Hypergraphs

Computer Science Department, Sapienza University of Rome, 00198 Rome, Italy

Received 4 October 2012; Accepted 23 October 2012

Academic Editors: E. Manstavicius and W. Menasco

Copyright © 2013 Francesco M. Malvestuto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Given a connected hypergraph with vertex set *V*, a *convexity space* on is a subset
of the powerset of *V* that contains *∅*, *V*, and the singletons; furthermore, is closed under intersection and every set in is connected in . The members of are called *convex sets*. The *convex hull* of a subset *X* of *V* is the smallest convex set containing *X*. By a *cluster* of we mean any nonempty subset of *V* in which every two vertices are separated by no convex set. We say that a convexity space on is *decomposable* if it satisfies the following three axioms: (i) the maximal clusters of form an acyclic hypergraph, (ii) every maximal cluster of is a convex set, and (iii) for every nonempty vertex set *X*, a vertex does not belong to the convex hull of *X* if and only if it is separated from *X* by a convex cluster. We prove that a decomposable convexity space on is fully specified by the maximal clusters of in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as “monophonic” and “canonical” convexities in hypergraphs and “all-paths” convexity in graphs).