Convexity Notions in Graphs

Let V be a finite set andM a finite collection of subsets of V . ThenM 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 or a convexity space. If S ⊆ V , then the convex hull of S, denoted by CH(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. This property is referred to as the Minkowski-Krein-Milman property. The distance between a pair of vertices u, v of G is the number of edges in a u − v geodesic (i.e., a shortest u − v path) in G and is denoted by dG(u, v) or d(u, v). The interval between a pair u, v of vertices in a graph G, denoted by IG[u, v] or I[u, v], is the collection of all vertices that lie on some u−v geodesic in G. A u−v geodesic is necessarily an induced path. The two most commonly studied convexity notions in graphs are defined in terms of shortest and induced paths between pairs of vertices. A subset S of vertices of a graph is said to be g−convex if it contains the interval between every pair of vertices in S and it is m−convex if for all pairs u, v of vertices in S it contains all vertices that belong to some induced u− v path. It is not difficult to see that the collection of all g−convex sets is an alignment of V as is the collection of all m−convex sets. A vertex in a graph is simplicial if its neighbourhood induces a complete subgraph. It can readily be seen that v is an extreme point for a g−convex or m−convex set S if and only if v is simplicial in the subgraph induced by S. It is true, in general, that the convex hull of the extreme points of a convex set S is contained in S, but equality holds only in special cases. In [8] it is shown that the g−convex sets of a graph G have the Minkowski-Krein-Milman property if and only if G is chordal and has no induced 3-fan; these are precisely the ptolemaic graphs. In the same paper chordal graphs are