  Let V be a finite set and M a collection of subsets of V. Then M is an
alignment of V if and only if M is closed under taking intersections and
contains both V and the empty set. If M is an alignment of V, then the elements
of M are called convex sets and the pair (V, M) is called an aligned space. If
S is a subset of V, then the convex hull of S is the smallest convex set that
contains S. Suppose X in M. Then x in X is an extreme point for X if X-x is in
M. The collection of all extreme points of X is denoted by ex(X). A convex
geometry on a finite set is an aligned space with the additional property that
every convex set is the convex hull of its extreme points. Let G=(V,E) be a
connected graph and U a set of vertices of G. A subgraph T of G containing U is
a minimal U-tree if T is a tree and if every vertex of V(T)-U is a cut-vertex
of the subgraph induced by V(T). The monophonic interval of U is the collection
of all vertices of G that belong to some minimal U-tree. A set S of vertices in
a graph is m_k-convex if it contains the monophonic interval of every k-set of
vertices is S. A set of vertices S of a graph is m^3-convex if for every pair
u,v of vertices in S, the vertices on every induced path of length at least 3
are contained in S. A set S is m_3^3-convex if it is both m_3- and m^3- convex.
We show that if the m_3^3-convex sets form a convex geometry, then G is A-free.
