  Connectivity and diagnosability are two important parameters for the fault
tolerant of an interconnection network $G$. In 1996, F\`{a}brega and Fiol
proposed the $g$-extra connectivity of $G$. A subset of vertices $S$ is said to
be a \emph{cutset} if $G-S$ is not connected. A cutset $S$ is called an
\emph{$R_g$-cutset}, where $g$ is a non-negative integer, if every component of
$G-S$ has at least $g+1$ vertices. If $G$ has at least one $R_g$-cutset, the
\emph{$g$-extra connectivity} of $G$, denoted by $\kappa_g(G)$, is then defined
as the minimum cardinality over all $R_g$-cutsets of $G$. In this paper, we
first obtain the exact values of $g$-extra connectivity of some special graphs.
Next, we show that $1\leq \kappa_g(G)\leq n-2g-2$ for $0\leq g\leq \left\lfloor
\frac{n-3}{2}\right\rfloor$, and graphs with $\kappa_g(G)=1,2,3$ and trees with
$\kappa_g(T_n)=n-2g-2$ are characterized, respectively. In the end, we get the
three extremal results for the $g$-extra connectivity.
