  The ratio of the connected domination number, $\gamma_c$, and the domination
number, $\gamma$, is strictly bounded from above by 3. It was shown by
Zverovich that for every connected $(P_5,C_5)$-free graph, $\gamma_c = \gamma$.
  In this paper, we investigate the interdependence of $\gamma$ and $\gamma_c$
in the class of $(P_k,C_k)$-free graphs, for $k \ge 6$. We prove that for every
connected $(P_6,C_6)$-free graph, $\gamma_c \le \gamma + 1$ holds, and there is
a family of $(P_6,C_6)$-free graphs with arbitrarily large values of $\gamma$
attaining this bound. Moreover, for every connected $(P_8,C_8)$-free graph,
$\gamma_c / \gamma \le 2$, and there is a family of $(P_7,C_7)$-free graphs
with arbitrarily large values of $\gamma$ attaining this bound. In the class of
$(P_9,C_9)$-free graphs, the general bound $\gamma_c / \gamma \le 3$ is
asymptotically sharp.
