Decomposable Convexities in Graphs and Hypergraphs

Francesco M Malvestuto

doi:10.1155/2013/453808

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).

Highlights

A convexity space [1] over a nite nonempty set VV is a subset CC of the powerset of VV that contains ∅ and VV and is closed under intersection. e members of CC are called convex sets. e convex hull of a subset XX of VV, denoted by ⟨XXX, is the smallest convex set containing XX
If XX is a convex set, the subspace of CC induced by XX is the convexity space CCCCCC C {AA A AA A AA A AAA on the subhypergraph induced by XX
Let H be an acyclic hypergraph on VV, XX a nonempty proper subset of VV, and uu a vertex in VV VVV. e following holds: (i) a vertex uu of H belongs to the vertex set of GR(H, XXX if and only if uu is separated from XX by no partial edge of H; (ii) if uu is not a vertex of GR(H, XXX, there exists an edge of GR(H, XXX that separates uu from XX in H; (iii) if AA is an edge of GR(H, XXX, AA is the union of AAA XX with the neighborhoods of the connected components of H − AA whose vertex sets are not disjoint from XX

Summary

Introduction

A ( nite) convexity space [1] over a nite nonempty set VV is a subset CC of the powerset of VV that contains ∅ and VV and is closed under intersection. e members of CC are called convex sets. e convex hull of a subset XX of VV, denoted by ⟨XXX, is the smallest convex set containing XX. We show that mm-convexity and ap-convexity belong to a wide class of convexity spaces on hypergraphs, which we call “decomposable” and de ne by means of three axioms. To this end, we need to introduce the notion of a “cluster.”. Our main result is that if CC is a decomposable convexity space on H, CC enjoys the following two properties.

Preliminaries

The Cluster Hypergraph

Axioms 1 and 2

Decomposable Convexity Spaces

Examples

Closing Note