The Steiner diameter of a graph

The Steiner distance of a graph‎, ‎introduced by Chartrand‎, ‎Oellermann‎, ‎Tian and Zou in 1989‎, ‎is a natural generalization of the‎ ‎concept of classical graph distance‎. ‎For a connected graph G of‎ ‎order at least 2 and S?V(G)‎, ‎the Steiner‎ ‎distance d(S) among the vertices of S is the minimum size among‎ ‎all connected subgraphs whose vertex sets contain S‎. ‎Let n,k be‎ ‎two integers with 2?k?n‎. ‎Then the Steiner‎ ‎k-eccentricity ek(v) of a vertex v of G is defined by‎ ‎ek(v)=max{d(S)|S?V(G)‎,|S|=k‎,andv?S‎‎}‎. ‎Furthermore‎, ‎the Steiner k-diameter of G is‎ ‎sdiamk(G)=max{ek(v)|‎v?V(G)}‎. ‎In 2011‎, ‎Chartrand‎, ‎Okamoto and Zhang showed that k?1?sdiamk(G)?n?1‎. ‎In this‎ ‎paper‎, ‎graphs with sdiam3(G)=2,3,n?1 are characterized‎, ‎respectively‎. ‎We also consider the Nordhaus-Gaddum-type results for‎ ‎the parameter sdiamk(G)‎. ‎We determine sharp upper and lower‎ ‎bounds of sdiamk(G)+sdiamk(G¯¯¯¯) and sdiamk(G)?‎‎sdiamk(G¯¯¯¯) for a graph G of order n‎. ‎Some‎ ‎graph classes attaining these bounds are also given.