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Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be aninteger with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa(S)$ denote the maximum number $\ell$ of edge-disjoint trees$T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for every pair$i,j$ of distinct integers with $1\leq i,j\leq \ell$. A collection$\{T_1,T_2,...,T_\ell\}$ of trees in $G$ with this property is called aninternally disjoint set of trees connecting $S$. Chartrand et al. generalizedthe concept of connectivity as follows: The $k$-$connectivity$, denoted by$\kappa_k(G)$, of $G$ is defined by $\kappa_k(G)=$min$\{\kappa(S)\}$, where theminimum is taken over all $k$-subsets $S$ of $V(G)$. Thus$\kappa_2(G)=\kappa(G)$, where $\kappa(G)$ is the connectivity of $G$.  In general, the investigation of $\kappa_k(G)$ is very difficult. Wetherefore focus on the investigation on $\kappa_3(G)$ in this paper. We studythe relation between the connectivity and the 3-connectivity of a graph. Firstwe give sharp upper and lower bounds of $\kappa_3(G)$ for general graphs $G$,and construct two kinds of graphs which attain the upper and lower bound,respectively. We then show that if $G$ is a connected planar graph, then$\kappa(G)-1 \leq \kappa_3(G)\leq \kappa(G)$, and give some classes of graphswhich attain the bounds. In the end we show that the problem whether$\kappa(G)=\kappa_3(G)$ for a planar graph $G$ can be solved in polynomialtime.

Sharp bounds for the generalized connectivity $\kappa_3(G)$