Given a connected hypergraph

A (

Most convexities in graphs and hypergraphs have been stated in terms of “feasible” paths [

We note that in graphs monophonic convexity (

In this paper, we show that

Let

The cluster hypergraph of

Every edge of the cluster hypergraph of

For every nonempty vertex set

Our main result is that if

there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of

The rest of the paper is organized as follows. In Section

Let

A vertex is a

Two vertices are

A

A (

Let

Let

A vertex

In the next two subsections we deal with “separators” and “acyclic hypergraphs.”

Let

A hypergraph is

A more efficient algorithm to test acyclicity was given in [

Let

Let

For our purposes, we need a modification of Graham reduction when there is a set

the edges of

the MVSs of

The following lemma contains one more property of

Let

a vertex

if

if

Figure

Let

First of all, observe that the connected component of

Let

Let

Since every clique of

By Theorem

Let

Let

The next lemma is useful for the sequel.

Let

Let

The next result relates the MVSs of the cluster hypergraph of

Let

Let

Consider the (hyper)graph

In this section we state some properties of convexity spaces that satisfy Axioms

Let

Let

Let

The following are two properties of convexity spaces that satisfy both Axioms

Let

Since

Let

We first prove that (i) every MVCS of

Consider again the (hyper)graph

Recall that a convexity space

In the next two subsections, we shall prove the properties (D1 and D2) mentioned in the introduction.

Given any convexity space

Let

Let

Since in

At this point, we are in a position to prove property (D1) of decomposable convexity spaces.

Let

By Lemma

Theorem

Let

every edge of

By condition (a),

First of all, recall that for any subset

We begin with a few basic definitions and preliminary results. A vertex

In order to prove property (D2), we make use of the following two characterizations of convex geometries.

A convexity space

The next characterization of convex geometries is based on the notion of a “descending path,” which is defined as follows. Let

A convexity space

Let

Since

At this point, we are in a position to prove property (D2) of decomposable convexity spaces.

Let

In both cases, by hypothesis,

It is worth observing that, by Theorem

In this section we prove that

Let

If

If

In order to prove the decomposability of

Let

It is sufficient to prove that two vertices of

Let

Let

Let

An efficient algorithm for computing

If

If

In order to prove the decomposability of

Let

Let

Let

Let

Let

At this point, it is a mere exercise to prove that the cluster hypergraph of

Given a convexity space

Consider the (hyper)graph

It is easy to see that the sets