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 g 3 -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 g 3 -convex in a graph G , then G has no induced house nor twin C 4 , 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 g 3 -convex.