Superconnectivity of regular graphs with small diameter

A graph is superconnected, for short super-κκ, if all minimum vertex-cuts consist of the vertices adjacent with one vertex. In this paper we prove for any rr-regular graph of diameter DD and odd girth gg that if D≤g−2D≤g−2, then the graph is super-κκ when g≥5g≥5 and a complete graph otherwise.