On Minimal and Minimum Hull Sets

A graph convexity (G,C) is a graph G together with a collection C of subsets of V(G), called convex sets, such that ∅,V(G)∈C and C is closed under intersections. For a set U⊆V(G), the hull of U, denoted H(U), is the smallest convex set containing U. If H(U)=V(G), then U is a hull set of G. Motivated by the theory of well covered graphs, which investigates the relation between maximal and maximum independent sets of a graph, we study the relation between minimal and minimum hull sets. We concentrate on the P3 convexity, where convex sets are closed under adding common neighbors of their elements.