A note on Cops and Robbers, independence number, domination number and diameter

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.) $c(G)\le \alpha (G)-\lfloor \frac{D(G)-3}{2}\rfloor$, and (ii.) $c(G)\le \gamma (G)-\frac{D(G)}{3}+O(\sqrt{D(G)})$.