Hamiltonicity of Domination Vertex-Critical Claw-Free Graphs

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 $3 \leq k \leq 5$.