  A graph $G$ is said to be $k$-$\gamma$-vertex critical if the domination
numbers $\gamma(G)$ of $G$ is $k$ and $\gamma(G - v) < k$ for any vertex $v$ of
$G$. Similarly, A graph $G$ is said to be $k$-$\gamma_{c}$-vertex critical if
the connected domination numbers $\gamma_{c}(G)$ of $G$ is $k$ and
$\gamma_{c}(G - v) < k$ for any vertex $v$ of $G$. The problem of interest is
to determine whether or not $2$-connected $k$-$\gamma$-vertex critical graphs
are Hamiltonian. In this paper, for all $k \geq 3$, we provide a $2$-connected
$k$-$\gamma$-vertex critical graph which is non-Hamiltonian. We prove that
every $2$-connected $3$-$\gamma$-vertex critical claw-free graph is Hamiltonian
and the condition claw-free is necessary. For $k$-$\gamma_{c}$-vertex critical
graphs, we present a new method to prove that every $2$-connected
$3$-$\gamma_{c}$-vertex critical claw-free graph is Hamiltonian. Moreover, for
$4 \leq k \leq 5$, we prove that every $3$-connected $k$-$\gamma_{c}$-vertex
critical claw-free graph is Hamiltonian. We show that the condition claw-free
is necessary by giving $k$-$\gamma_{c}$-vertex critical non-Hamiltonian graphs
containing a claw as an induced subgraph for $3 \leq k \leq 5$.
