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