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 P 3 convexity, where convex sets are closed under adding common neighbors of their elements.