Separation properties of 3-Steiner and 3-monophonic convexity in graphs

Let V be a finite set and C a collection of subsets of V . The ordered pair ( V , C ) is an alignment if C is closed under taking intersections and contains both 0̸ and V . If ( V , C ) is an alignment, then C is a convexity for V , and the elements of C are referred to as the convex sets of the convexity C . A convex set A is a half-space if V − A is convex. The following separation properties have been defined for a given convexity C of V . ( S 1 ) ery x ∈ V , the set { x } is convex. ) ery pair a , b ∈ V , there exist complementary half-spaces A , B such that a ∈ A and b ∈ B . ) ery convex set A and b ∈ V − A , there exist complementary half-spaces A ′ , B ′ in C such that A ⊆ A ′ and b ∈ B ′ . ) ery pair A , B ∈ C of disjoint convex sets, there exist complementary half-spaces A ′ , B ′ in C such that A ⊆ A ′ and B ⊆ B ′ . ell-known graph convexities satisfy property S 1 . Properties of graphs satisfying separation properties S 2 , S 3 , and S 4 with respect to the two most well-known graph convexities, namely, the geodesic and monophonic convexities, have been studied. In this paper we establish properties of graphs satisfying separation properties S 2 , S 3 , and S 4 relative to the 3-Steiner convexity and the 3-monophonic convexity of a graph.