Minimal trees and monophonic convexity

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 alignment or a convexity. If S ⊆ V , then the convex hull of S is the smallest convex set that contains S. Suppose X ∈ M. Then x ∈ X is an extreme point for X if X \ {x} ∈ M. 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. Several graph convexities are defined using minimal U -trees and structural characterizations of graph classes for which the corresponding collection of convex sets is a convex geometry are characterized.