  We study relations between diameter $D(G)$, domination number $\gamma(G)$,
independence number $\alpha(G)$ and cop number $c(G)$ of a connected graph $G$,
showing (i.) $c(G) \leq \alpha(G)-\lfloor \frac{D(G)-3}{2} \rfloor$, and (ii.)
$c(G) \leq \gamma (G) - \frac{D(G)}{3} + O (\sqrt{D(G)})$.
