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A graph \( G \) is said to be \( k \)-\(\gamma\)-edge critical if the domination number \(\gamma(G) = k\) and \(\gamma(G + uv) < k\) for every \( uv \notin E(G) \). For the connected domination number \(\gamma_c(G) = k\), the total domination number \(\gamma_t(G) = k\) and the independent domination number \( i(G) = k \), a \( k \)-\(\gamma_c\)-edge critical graph, a \( k \)-\(\gamma_t\)-edge critical graph and a \( k \)-\(i\)-edge critical graph are similarly defined. In our previous work, we proved that every \( 2 \)-connected \( k \)-\(\gamma_c\)-edge critical graph is hamiltonian for \( 1 \leq k \leq 3 \) and we provided a class of \( l \)-connected \( k \)-\(\gamma_c\)-edge critical non-hamiltonian graphs for \( k \geq 4 \) and \( 2 \leq l \leq \frac{n-3}{k-1} \). The problem of interest is to determine a sufficient condition for \( k \)-\(\gamma_c\)-edge critical graphs to be hamiltonian for \( k \geq 4 \). In this paper, we prove that every \( 2 \)-connected \( 4 \)-\(\gamma_c\)-edge critical claw-free graph is hamiltonian. For \( k \geq 5 \), we provide a class of \( k \)-\(\gamma_c\)-edge critical claw-free non-hamiltonian graphs of connectivity two. We further show that all \( 3 \)-connected \( k \)-\(\gamma_c\)-edge critical claw-free graphs are hamiltonian for \( 1 \leq k \leq 6 \). Our methodology also establishes some results on the hamiltonian properties of \( 3 \)-connected \( k \)-\(\mathcal{D} \)-edge critical claw-free graphs where \( \mathcal{D} \in \{ \gamma, \gamma_t, i \} \).