Graphs with large generalized 3-connectivity

Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,\mathrm{.}\mathrm{.}\mathrm{.},T_{\mathrm{\ell }}$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)=\mathrm{\varnothing }$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct integers $i,j$, where $1\le i,j\le r$. For an integer $k$ with $2\le k\le n$, the generalized $k$-connectivity ${\kappa }_k(G)$ of $G$ is the greatest positive integer $r$ such that $G$ contains at least $r$ internally disjoint trees connecting $S$ for any set $S$ of $k$ vertices of $G$. Obviously, ${\kappa }_2(G)$ is the connectivity of $G$. In this paper, sharp upper and lower bounds of ${\kappa }_3(G)$ are given for a connected graph $G$ of order $n$, that is, $1\le {\kappa }_3(G)\le n-2$. Graphs of order $n$ such that ${\kappa }_3(G)=n-2,n-3$ are characterized, respectively.