On 3-Steiner simplicial orderings

Let G be a connected graph and S a nonempty set of vertices of G . A Steiner tree for S is a connected subgraph of G containing S that has a minimum number of edges. The Steiner interval for S is the collection of all vertices in G that belong to some Steiner tree for S . Let k ≥ 2 be an integer. A set X of vertices of G is k -Steiner convex if it contains the Steiner interval of every set of k vertices in X . A vertex x ∈ X is an extreme vertex of X if X ∖ { x } is also k -Steiner convex. We call such vertices k -Steiner simplicial vertices. We characterize vertices that are 3-Steiner simplicial and give characterizations of two classes of graphs, namely the class of graphs for which every ordering produced by Lexicographic Breadth First Search is a 3-Steiner simplicial ordering and the class for which every ordering of every induced subgraph produced by Maximum Cardinality Search is a 3-Steiner simplicial ordering.