On a local 3-Steiner convexity

Given a graph G and a set of vertices W⊂V(G), the Steiner interval of W is the set of vertices that lie on some Steiner tree with respect to W. A set U⊂V(G) is called g3-convex in G, if the Steiner interval with respect to any three vertices from U lies entirely in U. Henning et al. (2009) [5] proved that if every j-ball for all j≥1 is g3-convex in a graph G, then G has no induced house nor twin C4, and every cycle in G of length at least six is well-bridged. In this paper we show that the converse of this theorem is true, thus characterizing the graphs in which all balls are g3-convex.