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 $S\subseteq V(G)$,the\mathit{\text{Steiner}}\mathit{\text{distance}}$d(S)$amongtheverticesof$S$ is the minimum size among all connected subgraphs whose vertex sets contain $S$. Let $n,k$ be two integers with $2\le k\le n$. Then the \emph{Steiner $k$-eccentricity $e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max {d(S),|,S\subseteq V(G), \ |S|=k, \ and \ v\in S }$. Furthermore, the \emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max {e_k(v),|, v\in V(G)}$\mathrm{.}In2011,Chartrand,OkamotoandZhangshowedthat$k-1\leq sdiam_k(G)\leq n-1$\mathrm{.}Inthispaper,graphswith$sdiam_3(G)=2,3,n-1$ are characterized, respectively. We also consider the Nordhaus-Gaddum-type results for the parameter $sdia{m}_k(G)$. We determine sharp upper and lower bounds of $sdia{m}_k(G)+sdia{m}_k(\bar{G})$ and $sdia{m}_k(G)\cdot sdia{m}_k(\bar{G})$ for a graph $G$ of order $n$. Some graph classes attaining these bounds are also given.