Graphs with 4-steiner Convex Balls

Recently a new graph convexity was introduced, arising from Steiner intervals in graphs that are a natural generalization of geodesic intervals. The Steiner tree of a set W on k vertices in a connected graph G is a tree with the smallest number of edges in G that contains all vertices of W . The Steiner interval I(W ) of W consists of all vertices in G that lie on some Steiner tree with respect to W . Moreover, a set S of vertices in a graph G is k-Steiner convex, denoted gk-convex, if the Steiner interval I(W ) of every set W on k vertices is contained in S. In this paper we consider two types of local convexities. In particular, for every k > 3, we characterize graphs with gk-convex closed neighborhoods around all vertices of the graph. Then we follow with a characterization of graphs with g4-convex closed neighborhoods around all g4-convex sets of the graph.