The Steiner diameter of a graph

Abstract: 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 \emph{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 $2\leq k\leq n$. 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 $sdiam_k(G)=\max {e_k(v)\,|\, v\in V(G)}$. In 2011, Chartrand, Okamoto and Zhang showed that $k-1\leq sdiam_k(G)\leq n-1$. In this paper, graphs with $sdiam_3(G)=2,3,n-1$ are characterized, respectively. We also consider the Nordhaus-Gaddum-type results for the parameter $sdiam_k(G)$. We determine sharp upper and lower bounds of $sdiam_k(G)+sdiam_k(\overline{G})$ and $sdiam_k(G)\cdot sdiam_k(\overline{G})$ for a graph $G$ of order $n$. Some graph classes attaining these bounds are also given.